Lecture Notes in Physics Edited by H. Araki, Kyoto, J. Ehlers, M(Jnchen, K. Hepp, Z~rich R. Kippenhahn, MLinchen, H. A. Weidenmeller, Heidelberg and J. Zittartz, K61n
190 Karl Kraus
States, Effects, and Operations Fundamental Notions of Quantum Theory Lectures in Mathematical Physics at the University of Texas at Austin Edited by A. BShm, J. D. Dollard and W. H. Wootters
Springer-Verlag Berlin Heidelberg New York Tokyo 1983
Author
Karl Kraus Physikalisches Institut der Universit~t WGrzburg Am Hubland, D-8700 WGrzburg Editors
A. B0hm Physics Department, University of Texas Austin, TX 78712, USA J.D. Dollard Mathematics Department, University of Texas Austin, TX 78712, USA W. H. Wootters Department of Physics and Astronomy, Williams College Williamstown, Mass. 01267, USA
ISBN 3-540-12732-1 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12732-1 Springer-Verlag New York Heidelberg Berlin Tokyo
Library of Congress Cataloging in Publication Data. Kraus, Karl, 1938- States effects, and operations.(Lecture notes in physics; 190) 1. Quantumtheory. I. Title. II. Series.QC174.12.K72 1983 5303'2 83-16906 ISBN 0-387-12732-1 (U.S.) This work is subject to copyright.All rights are reserved,whether the whole or partof the material is concerned,specifically those of translation,reprinting, re-useof illustrations,broadcasting, reproduction by photocopying machineor similar means,and storage in data banks. Under § 54 of the GermanCopyright Law where copies are madefor other than private use, a fee is payableto "VerwertungsgesellschaftWort", Munich. © by Springer-VerlagBerlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2153/3140-543210
Dem Andenken meiner Mutter
Marie
Kraus
geb.
Paus
4. Juli 1915 - 13. Oktober
1980
Preface
The
lecture notes
of
this volume are based
on a series
of lectures
given in the mathematical physics program at the University of Texas at Austin. of and
These lecture series were started in 1976 with the purpose
establishing of
communication
informing graduate
between mathematicians
students
and
of both departments
physicists
about
recent
developments in mathematics and physics. The lectures are directed at "non-speclallsts",
and
this
volume
also
should
be
suitable
for
a
general audience.
The subject of this volume is the interpretation and the foundations of quantum theory. There exist diverging opinions about this subject. Some people believe that quantum physics is so remote from the usual way of thinking and so foreign to everyday language that its meaning cannot
be communicated
directly.
According
an understanding of quantum mechanics
to them,
the only way to
is to study many separate pro-
blems and thereby acquire an intuitive feeling for the foundations of quantum mechanics.
The
premise
for
the work
the foundations of quantum mechanics
discussed
can be verbalized,
here
is that
and the pre-
sent lectures are an attempt to communicate them.
But
the
champions
concerning
the
of
this field also have
interpretation
of
different
quantum mechanics.
points The
of view
purpose
of
these lecture notes is not to give a comparative review of the different approaches -
this can be found in "Interpretations and Founda-
tions
Theory",
(1981). Ludwig
of
Quantum
Here
just
and his
encompass most
one
school,
H.
approach
Neumann is
BI-Wissenschaftsverlag
presented,
w h i c h appears
of the other orthodox
quantum mechanics.
(Ed.),
to be
the
developed
by
sufficiently general
to
(Copenhagen)
one
interpretations of
Vl
The usual
custom
postulates them
the
a very
usual
operators to only
in
in
in this field
formalism
linear
road
to
with
the
tors
in a Hilbert
this
the
mathematically
fundamental
This
is the road
make
this
Arno B~hm J.D. Dollard W.H. Wootters (Editors)
mathematical quantum
product
more
statements
mechanics
spaces.
of
familiar
rather
in
Even
and
derive
terms
of
quantum
mechanics
objects
of
and
in these lectures
subject
accessible
linear
A more
is to start
operators
them,
from
oneself
procedure.
in terms of these mathematical
fundamental
meaningful
restricting
tedious
than deriving
that has been followed and
language,
is still a very
foundations
space,
important
audience.
of
scalar
one approach,
accessible
the
general
is to start from physically
and
vec-
to formulate quantities. in order to to
a
larger
Author's
The subject
of these notes
fically,
they
structure
of
extent
deal that
when
writing
with
the
standard
also
with
some
these
- at
notes,
in the general
structure
and
the
reader
I had in
already
be
and
physical
the physical
conceptual to this
mechanics,
to specific
More speci-
form and
The
should
quantum
in applications
of
in this
textbooks.
therefore, of
mechanics.
aspects
least
in the usual
formulation
not only
quantum
particular
which
treated
mind
interested
is ordinary
theory,
- are not
Preface
familiar should
problems
interpretation
be but
of the
theory.
As
quantum
mechanics
linear operators matical cal
suitable ding.
are
has
required, if
the same
reason,
full
detail.
in
such
technicalities
These notes Texas
at
Austin
topics
discussed
dings
of
the
on these matters
results
in
an
Spaces"
occasion
several
selfcontalned
readers
will
just
way.
to
in
them
at
who
not
over
including rea-
are presented
too much
them,
the
and
interested concentrate
has
in
given at the University
An
earlier
been
the University
to
rethink
account
published of
1973
at
to
-
statements.
1980.
Marburg
mathemati-
facilitate
deductions
are
"Foundations
aspects, and
to
and
of this mathe-
explain
of lectures
lectures on
of vectors
advanced
order
glance
September
conference
terms
more
most mathematical
in these
lecture
with
-
that
in
Linear
me
tried
on a series
Ordered
ded
have
Whenever
on the more essential
are based
in
some basic knowledge
necessary
I hope
their attention
space,
I
references,
For
formulated
been assumed.
in
of
usually
in Hilbert
framework
tools
is
Quantum [I].
The
of Texas
the whole reformulate
the
in the proceeMechanics invitation
at Austin
subject,
of
to
everything
and to
provi-
extend
my
in a more
VIII
For
this
invitation,
sant stay at Austin, Griffy, ters my
Dr.
for their kind hospitality
English,
I want
W.
I am also very grateful
who read a preliminary version
and
suggested
several
other
to Dr. W.H. Woot-
of my notes,
improvements.
corrected
L a s t but not
to express my hearty thanks to Mrs. Ch. Steinbauer and
Petzold
(WUrzburg),
who
have
task of typewrinting and proofreading, Springer
during my plea-
I want to thank Profs. A. Bohm, J. Dollard, T.A.
and J.A. Wheeler.
(Austin),
least,
and
Verlag
for
his
great
been
burdened
with
the tedious
and to Prof. W. Beiglb~ck from
patience
and
during the preparation of the final typescript.
constant
encouragement
Contents
§I States and Effects §2 Operations
. . . . . . . . . . . . . . . . . . . . .
1
. . . . . . . . . . . . . . . . . . . . . . . . .
13
§3 The First R e p r e s e n t a t i o n T h e o r e m
§4 Composite Systems
. . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
42
62
§5 The Second R e p r e s e n t a t i o n T h e o r e m
. . . . . . . . . . . . .
81
§6 Coexistent Effects and Observables
. . . . . . . . . . . . .
103
References
. . . . . . . . . . . . . . . . . . . . . . . . . .
150
§i States and Effects
Almost
all treatments
of quantum mechanics
mental
importance
the
physical
to
interpretation
notions
these
different
investigation,
"state"
of these notions,
bly from one author to another. of
of
points
in view
agree
and
however,
funda-
"observable". differs
The
considera-
The discussion of as many as possible
of view is not
of
in ascribing
both
the
the subject
limited
space
of the present
available
and
the
insufficient familiarity of the author with most of them.
We prefer
instead
to follow here as closely as possible one particu-
lar interpretation of quantum mechanics, Ludwig be
and his
school
particularly
theory would We shall
which has been elaborated by
[2], and which the present author considers
satisfactory.
A
still go far beyond
detailed
exposition
the limits
is interested
Ludwig's
of space available here.
try to sketch some of the main ideas only,
any reader who
of
to
in more details
and to encourage
to consult
the original
literature quoted above.
Another
reason
interpretations technical
for
not
entering
a
detailed
of quantum mechanics
parts
of
the
present
discussion
is our conviction
work,
if
suitably
of
that
various the more
reformulated,
are
compatible with most or even all of them. We thus believe that even a reader who disagrees many
profit
completely with the interpretation
from the following
reinterpreting
our
results
in
investigations, terms
of
his
proposed here
even if the burden of
preferred
interpretation
of quantum mechanics is entirely left to himself.
According
to
Ludwig,
any
interpreted "from outside",
physical
theory
is
in
some
sense
to
be
i.e., in terms of "pretheories" not belon-
ging to the theory in question itself. For quantum mechanics in particular, and
these
describe
"pretheories" the
belong
construction
and
to the
realm of classical
application
physics,
of macroscopic
prepa-
2 •
ring
and
measuring
instruments.
advocated
by
who
classical
nature
Bohr, of
A
similar
repreatedly
measuring
point
stressed
instruments
of
the
for
view
has
importance
the
been
of
the
understanding
of
quantum mechanics.
Such
statements
vior
of
should
macroscopic
If
invented. ticular
this
theories
were
Indeed,
be misinterpreted
instruments
in terms of classical namics.
not
true,
can
be
that
quantum
mechanics
of macroscopic
just
would
serves
instruments
the beha-
understood
such as, e.g., mechanics
quantum mechanics
behavior
always
to mean
completely
and electrody-
never
have
to describe
which
been
some par-
can not
be explai-
ned classically.
It
is maintained,
instruments purely
can
during
ter,
are
"naive"
of
quantum
"objective
this
a
Ludwig's ties,
and
respect
events. theory be
radical
changes
theless,
this
cover,
such
changes
as,
objectively
and
always
reference
typical
mechanics
descisive
can
any
e.g.,
real
experimentalists
finally,
to
(at
the
r01e It
in our "naive"
and
are to
application
of
described
in
-
quantum
occurlng
mechanics.
in such instru-
the discharge
e~ents,
always
in
accept
is
of a coun-
much such
the
that
other
occurrences
one
quantum
of
and
interpreted
everyday
concept
concept
has
"things"
as atoms
the
mechanics,
the
with
refined
or electrons.
mechanics ascribing,
creation
of
motivations
of
all its subtle-
consistently
be
of
surrounding
quantum
main
of physical
to
would
in
interpre-
on ascribing
interpretations,
consciousness
fact
of
same
in terms
rests
macroscopic
interpretation
human
entirely
ultimately
the
certain
in
formulated
thus
least)
to
formulated
then
present
with
to show
such
is
events,
disagrees
observable
construction
practice,
without
as
reality"
profoundly
the
and as we all do in everyday life. The physical
instruments
e.g.,
in
"measurements",
in which
in practice,
In
and,
terms,
accepted
way
tation
-
that
in the same spirit,
ments
us.
be
classical
Moreover,
such
however,
without
reality.
such
(Never-
considerably
to
This problem has
also
been
analyzed
by Ludwig,
but
the results
of this deep analysis
cannot even be sketched here.)
According of
to the point
quantum mechanics
of macroscopic preliminary preparing
of view adopted here,
have
instruments
notion
of
thus
instruments.
to be defined
and prescriptions
"state"
the fundamental notions
tells
us
in terms
for their application.
is then most
Experience
operationally
simply given that
in terms
suitably
A of
constructed
instruments can be used to produce ensembles - in principle arbitrarily
large - of
(e.g.,
electrons).
ensemble hand
single microsystems
is then,
notation
for
Ascribing
this
specified
at this lowest the
applied
purpose,
the
notion
the
technical
the
entire preparation
stems
of
are
thus
a
"state"
a
"prestate". of
A
the
accordingly,
prestates,
such
an
We
introduce,
prestate
preparing
is
thus
instrument
is abbreviated here
the same label w shall
the applied preparing procedure - itself.
to
just a short-
procedure.
Such a specification
in different
type considered
level of the theory,
preparation
labels w for prestates;
also be used to denote
particular
like
description
and its mode of application. by using
the
something
for
by
of
instrument - or rather,
Two ensembles
w I $ w2,
of mlcrosy-
if and only
if they
are produced by different preparing procedures.
Another empirical fact is the existence of so called measuring instruments,
which
are
capable
of
undergoing
macroscopically
observable
changes due to ("triggered by") their interaction with single microsystems.
The simplest type of measuring
a single change may be triggered.
instrument
For instance,
is one on which just an originally charged
counter may be found either still charged or discharged,
a~ter it has
been exposed to an electron emitted by some preparing apparatus. result will depend, ter,
and
on whether
loosely or not
speaking, the
on the efficiency
electron
"hits"
it.)
(The
of the coun-
Instruments
of
this type perform so called yes-no measurements:
calling the observa-
ble
usually
change
result
of
the
instrument
of a single measurement
an
"effect",
one
defines
the
to be "yes" if the effect occurs,
and
"no"
if
the
effect
and sometimes rence
of
does
not
occur.
even appropriate,
the effect,
in this way,
and
to associate
vice
the apparatus
It is equally
versa.
(With
then performs
possible,
however,
"yes" with
the non-occur-
its reading
reinterpreted
a different - although close-
ly related - yes-no measurement.)
For
many
values" this
purposes
1 and
0 with
convention,
measurements,
notion
considered nics. can
It be
which
of we
results
also
in
a
The latter the
fit
that
with,
thus
a
broader
e.g.,
class
a movable
instruments
with
them
facts
as
not only
elementary
With of
poin-
more than only two possible
basic
way,
"measured
respectively.
a
- and
need
the
not
quantum
be
mecha-
of any observable
combination
simple
building
-
of
the measurement
standard
are
into
complicated
the
associate
"no",
connected
discuss known
to
and
contains
such more
well
also
"yes"
instruments
in general
first
interpreted,
but
convenient
"observables"
is indeed
measurements.
more
measurements
However,
when
urements,
the
involving
values.
general
is
yes-no
ter on a scale, measured
it
of
prototypes
blocks
for
yes-no
of meas-
more
general
ones. We shall return to this point in §6.
An instrument ratus, As
in
and the
complete
which
shall case
also of
technical
structions
Assume
performing
now then
yes-no measurements be
symbolized
preparing
description
preparing
interacts
the
an or
apparatus
f.
effect
on
assume
such
times.
(Keeping w and f fixed means,
single
identically
the
this
letter,
label
apparatus,
produces apparatus
non-occurrence
Call
experiments,
constructed
w
effect
to
the
occurrence
some
f
usually
stands
including
f.
for
the
a
in-
and its reading.
either
least
the
of
instrument
with
by
instruments,
for its application
a
here
is called an effect appa-
with
this
a
a
single f, of
"single
microsystem,
leading the
in
turn
corresponding
experiment",
and
given w and f, to be repeated of course,
instruments
N
to use the same or at
in all single
experiments.)
We may
then also say that the effect apparatus
f has been applied to
an ensemble of N microsystems in the prestate w.
The
effect
apparatus
experiments, N+ will ment
f
will
not
Nevertheless,
be
determined
large,
large N.
triggered stems
answer
"yes"
in
N+
i.e.,
single
In general,
the outcome of a single experi-
completely
by
the
instruments
w and
f.
in each series of N single experiments with given w and
f the fraction N+/N comes
very
the
and "no" in the remaining N_ = N - N+ cases.
be neither N nor zero;
will
ciently
yield
out roughly
the same,
such that N+/N approaches Thus experience
with
prepared
reproducible
a definite
is suffi-
limit ~(f,w)
tells us that effect apparatuses relative
in a prestate w.
N+/N from ~(f,w)
if only N
frequencies
Deviations
~(f,w)
by
for
f are
mlcrosy-
of the observed values o f
can then be interpreted as statistical errors due to
the finiteness of N.
This
most
laws
of quantum mechanics.
we
call
~(f,w)
apparatus occurs be
crucial
the
the
particular
case
of
the
as
In accordance
w.
for
synonymous
measured
with
just values
basis with
the
Whenever
interpretation
of ~(f,w)
experiments,
empirical
"probability"
in the present
average
is
f in the prestate
understood
single
fact
the
the usual
triggering
statistical terminology,
of
the
of quantum mechanics, frequency",
considered. (yes)
effect
the notion of "probability"
"relative
1
for
Since
and
we may also call ~(f,w)
0
N+/N
it has to as
is
in
the
also
the
(no) o b t a i n e d
in
N
the expectation value of
the f measurement in the prestate w.
Every
experimentalist
preparing
knows
that
some
instrument w or an effect
minor
apparatus
out affecting their statistical behavior, lity
function
intruments
~(f,w).
But
even
two
technical
details
of
a
f may be changed with-
as expressed by the probabi-
completely
different
preparing
w I and w 2 may give rise to the same probabilities
triggering of arbitrary effect apparatuses f; i.e.,
for the
~(f,w I) = ~(f,w2) for all f. Likewise,
there
certainly exist
(slightly,
(1.1)
or even completely,
diffe-
rent) effect apparatuses fl and f2' such that ~(fl,w) = ~(f2,w) for all w; i.e.,
fl and
f2 are
triggered
with
equal
(1.2)
probabilities
in arbitrary
prestates w. If, as usual, we restrict our attention to the probabilities ~(f,w),
as the basic quantities for the formulation of the stati-
stical
of
laws
preparing
instruments
probabilities w2
satisfying
and f,
F
then
inessential.
(1.1),
as
Eqs.
preparing
respectively.
usual,
differences
well
Accordingly,
as
two
(I.i)
and
instruments
All
(1.2)
two
effect
between
instruments
prestates
apparatuses
w
two
these
w I and
fl
and
f2
If thus read as equi-
define
(prestates)
equivalence and
effect
classes W
apparatuses
w or f in a given equivalence
W or F thus behave "statistically", ~(f,w),
any
(1.2), are defined to be equivalent.
relations, of
mechanics,
or effect apparatuses which do not affect
become
satisfying
valence
quantum
class
with respect to the probabilities
in the same way. An equivalence class W is called a state, as whereas - in accordance with Ludwig's
equivalence
class
red
instrument
by
an
ensemble
F an effect. w
in
An ensemble
the
terminology - we call an of N microsystems,
equivalence
class
W,
is
prepa-
called
an
in the state W, whereas an effect apparatus f in the equiva-
lence class F is said to measure the effect F.
By definition of these equivalence classes, rise
to a new function ~(F,W),
the function ~(f,w)
called the probability
gives
for the occur-
rence of the effect F in the state W, as defined by
~(F,W) = 7(f,w) with
f e F
and
w e W.
By
(i.i)
and
depend
on the choice of particular
on
right
the
hand
side
of
(1.3).
(1.2),
(i.3) this
definition
representatives Denoting
the
does not
f of F and w of W
sets of states W and
effects
F
by K and L,
on L × K.
According
respectively,
(1.3)
to its physical
thus
meaning,
defines
this
a function
function
obviously
satisfies 0 ~ ~(F,W) for
all
F e L
and
all
W s K.
~ 1
(1.4)
Moreover,
(1.1) and (1.2) of the equivalence
(1.3)
and
the
definitions
classes W and F also imply that
W 1 = W 2 iff ~(F,W I) = ~(F,W2)
for all F g L ,
(1.5)
F I = F 2 iff ~(FI,W)
for all W e K .
(1.6)
and
This
means,
W 1 @ W2, lead
in
particular,
as well
as
= ~(F2,W)
that
apparatuses
to a different
ensembles
measuring
"statistics",
in
different
and can thus
different
states
effects
F 1 @ F2,
be distinguished
expe-
rimentally.
One
of
the main
mathematical cally
goals
structure
meaningful
function
~.
(The
is no room here the
shall
thus
tions
of
operators
on
state
our
attention
electrons),
as based
may
space
space
of
to
we
the
here
sufficiently
shall also
formalism that
an
"axiomatic"
of
quantum
(ii) projection
approach
quantities
mathematically
Hilbert
space By
systems
possible
H,
in
restricting, (like,
complications
which
and
we
terms we
of
call
moreover,
e.g.,
by density operators W on H, effects,
than
and rela-
which
rules.
operators E on H represent
there
mechanics,
we shall assume - as usual - that
(i) states may be represented
the
physi-
(See [2].) Rather
the basic
"simple"
avoid
and
and second law.) Again
considered.
se could arise from superselection
Accordingly,
such
represented
system
suitable
of
sets K and L and the probability for
separable)
the derivation
from
this in any detail.
be
(complex,
the
on the first
Hilbert
theory
is
mechanics
example
for granted
a
the
for
to discuss
take
approach
quantum
classical
usual
the
of
postulates
is thermodynamics,
deriving
of Ludwig's
single otherwi-
(iii)
the
probability
for
the
occurrence
of
such
an
effect
E
in
a
state W is given by ~(E,W) with tr denoting
For
simplicity
rent
symbols
tors
on H
and
on
reas stand
for
right
(iii)
for
classes
of notation, states
hand
and
the
on
effects
on
on
other
them
the
side
of
(1.7),
the
left
hand
corresponding
,
(1.7)
trace.
we have avoided
and
representing
the
in
the operator
= tr(EW)
the introduction
the
one hand.
W and E side
physical
hand,
of
and
the
in
(i),
Thus,
denote (1.7)
quantities
of diffeopera(ii),
operators, the
same
(i.e.,
whe-
letters
equivalence
of instruments).
Statement
(i)
shall
corresponds
a
assumption,
the
of density
mean,
unique
more
precisely,
density
set K
operators
of
operator,
states
on H,
i.e.,
may
be
that and
to
vice
identified
of non-negative
every
state
versa. with
there
With
the
(and thus)
this
set K(H) Hermitean
operators W with unit trace: W = W* ~ 0 ,
K(H)
is
a subset
of
the
trace
trW = 1 .
class
B(H)I,
(1.8)
consisting
of
all
opera-
tors T on H for which the trace norm
llTil1 : tr[(T*T) I/2] is
finite.
e.g., its
[3],
For
the
Ch.
i, or [4].
Hermitean
part
operators - K(H)
properties
B(H)~,
is convex;
of B(H) 1 -
As a subset
(1.9)
to be
used
later
on - see,
of B(H) I - or alternatively,
consisting
of
all
Hermitean
trace
belongs
state
mixing.
using
N 1 = %N
to
K(H).
class
i.e., with 0 < % < I and Wl, W 2 g K(H),
(I.I0)
W = %W 1 + (I-%)W 2
also
of
Physically,
this
expresses
the
possibility
of by
Assume
an ensemble
of N >> 1 systems
to be prepared
times
a preparing
instrument
N2 =
w I and
(1-%)N
times
another fixed, the
one,
w 2.
defines
prestates
effect
A
prescription
a new wI
apparatus
of
preparation
and f is
w2
with
applied
this
type,
procedure
relative to the
w,
with called
weights
ensemble,
prepared
~,
~ and
wI
and
w2
the mixing
of
1
it will
~.
be
If
triggered
~(f,Wl)N 1 times
by the subensemble
by the instrument
N(f,w2)N 2 times
by the N 2 systems
the probability
for the triggering of f in the mixed prestate w is
prepared
an
by the instrument
Wl, and w 2. Thus
~(f,w) : (~(f,Wl)N 1 + ~(f,w2)N2)/N
: lu(f,wl) + (l-l)~(f,w2) •
This
!
relation
w~ leads
implies
that
replacing
to a w' equivalent
holds true for equivalence
w I and w 2 by equivalent
to w, and that an analogous
in particular,
tors E, the last relation,
relation also
classes:
N(F,W) = I~(F,WI) + (I-%)~(F,W2)
If applied,
w I and
to effects
described
(I.Ii)
.
by projection
opera-
together with (1.7), leads to
tr(EW) = %tr(EW I) + (1-l)tr(EW 2)
= tr[E(~W I + (I-X)W2) ] . As the an
E
is
arbitrary,
projection arbitrary
this
operator
unit
onto
vector
A = W - IW I - (I-%)W 2.
implies
(I.i0).
(Take,
the one-dimensional
f e H.
Then
(1.12)
Since A is linear,
E = I f >< f l,
subspace
spanned
by
(f,Af) = 0, with
we also have
(f,Af) = 0 for
identity
- ((f-g),A(f-g))
+ i((f-ig),A(f-ig))
yields
e.g.,
becomes
vectors f of arbitrary length, and the polarization
4(f,Ag) = ((f+g),A(f+g))
(1.12)
- i((f+ig),A(f+ig))
(f,Ag) = 0 for all f, g s H, i.e., A = 0.)
(1.13)
10
A
state
W
satisfying
interpretation, not
proper
(i.i0)
a mixture
mixtures,
(i. I0)
with
state,
as
W 1%
called
of the states
i.e.,
W2
is thus
and
which
can
be
are
corresponds
of
this
W I and W 2. States
not
0 < X < i,
is well-known,
in view
to a
W which
represented
called
pure
physical
in
the
form
A
pure
states.
one-dimensional
are
projection
operator, W = if >< f l ,
and
is
usually
"state
vector"
represented
by
the
sufficient
satisfactory states
rid
to
only.
procedures of
For
state
for
many
restrict
like
procedures,
unit
f. In this case, Eq.
~(E,W)
Although
even
the
single
ray
{ei~f}
spanned
by
the
reads
it
discussion
instruments,
nevertheless
quantum
as
state
there
is
of
disregarding
since
(1.14)
= (f,Ef)
above-described
mixtures,
with
(1.7)
purposes,
the by
llflf = 1 ,
are
also
which
very
to
pure
states
"artificial" mixing,
not
one
preparation
would
"simple"
not
get
preparation
nevertheless
do
not
pro-
duce pure states.
Statement
(ii)
operator vice
E
versa,
operator modify
above
on H every
are
yes-no As we
assumption
on H as describing which
meant
here
an
effect.
describes
E on H. this
is
measurement shall
by
This
and
imply
Usually
that one
every
also
can be described
however,
admitting
effects.
non-negative
see,
to
there
a larger
set L(H)
(therefore)
assumes
that,
by a projection
are
good
set L(H)
consists
projection
reasons
to
of operators
F
of all operators
F
Hermitean,
and
bounded
from
above by the unit operator:
0 ~ F = F* < 1 .
The ment
set
of
(iii)
projection
operators
is then generalized
(1.15)
E is a proper
to arbitrary
subset
effects
of L(H).
State-
F by requiring
Ii
~(F,W)
(i.i6)
= tr(FW)
for all effects F e L ~ L(H) and all states W ~ K ~ K(H).
The set L(H) is also convex,
i.e., containing with F 1 and F 2 also
(i.i7)
F = ~F i + (i-l)F 2
for
all
real
%
corresponding gous
case of Eq.
"mixed" times
to
the
Since
"mixing"
of
the
projection
I.
an
fl'
and
the effect
still
hope
(1.17)
to projection
shall
try
to
projection
does
that at least
all
not
show
later
on,
"decision
the
convex
- called
subset,
the
is general and them
actual proper
about
tion operators)
to
above
be
as
%N
to be
convex,
such
in the
described
prescription
for
by
a
meas-
so that one might
"simple"
convex of
however,
operators effects"
to be realized set L(H)
states,
of pure
agreement
yield
statements
K(H)
set
that many -
unlikely
set
were
that
f2 has
effect,
the
effect
apparatuses
operators.
effects
the
means
not
if every
sufficiently
cular
respect,
the
is
look very natural,
projection
this
operators
f
fl and f2
Applying
the apparatus
mechanics,
On the contrary,
In
respectively:
be allowed
fied.
over are rather
F2,
times
indeed,
apparatus
as in the analo-
of N microsystems
(I-%)N
not
effect
two effect apparatuses
F 1 and
quantum Now,
particular
F could be obtained,
ensemble
of
would
of
A
by "mixing"
to
operator.
uring
correspond
and
effects
set
effects
formulation
we
f
apparatus
applied.
usual
(1.10),
apparatus the
0
to such an effect
corresponding
As
between
that
which
states.
mixtures
the particular
by Ludwig
of
effects
also
instruments
rSle
is not
justi-
only very
parti-
[2] - which more-
at all in actual
states
rather
hope
describe
than
experiments.
is quite
contains,
But whereas
the pure
preparing
this
similar
as a non-convex
in the latter
case
form only a subset - or perhaps pure
states,
of the decision
in L(H) are much less popular.
to
there
of K(H),
even most
of
corresponding
effects
(projec-
12
Pure states cannot
be
can be characterized prepared
then ask whether distinguished ding
discuss
proper
and how,
physically
to Ludwig
"most
as
sensitive"
this here,
state
mixtures
similarly, from more
[2], decision effects
operationally
in
suitable
effects
above).
One might
F in L(H).
Accor-
are indeed distinguished as the subsets
but will discover another
of decision effects in §6.
(see
the decision effects E could be
general
effects
by the fact that they
of L(H).
We
characteristic
shall not property
13
§2 Operations
We
shall
now
introduce,
cept
of an "operation".
also
be defined
an
preparing state
W.
N >> 1
instrument
w,
so that
moreover,
thus
available
the apparatus
and
for
but
hand,
many
instead the are
"non-destructively"
in
class,
the
stems and
ment w. This
equivalence This
instruments
many the
state,
or
new
con-
this concept
been
shall
theory by
prepared
by
a
is in the corresponding
apparatus is
f is applied
still
"
after
its
f,
which
do
discussion
effect above
present
-
to and
interaction
sense.
with
preparing class
with
the effect apparatus
depends
w,
which
instruments on
both
f constituting
the
which
act
F
the
be
equiva-
f.)
the N microsy-
f as a new ensemble,
state W,
effect then
can
that every effect F
which
the combination
given
To
On
effect
state W as prepared
the
instrument
well
to consider
the apparatus
a quantum
and
of preparing
clearly,
to expect
to considering w
apply.
f in the corresponding
it is allowed
ensemble
as
a given
this
microsystem.
not
f
satisfy
by a suitable apparatus
from the original
instrument
As
not
the
does
apparatuses
instruments
interaction
to this
"destroy"
even reasonable
just amounts
preparing new
microsystem
described,
their
to ascribe
single
each
"non-destructively"
will be different
nal
has
an effect
"absorb"
it appears
situation after
the
used in quantum
the ensemble
experiments
by many different
may be measured
In
first
microsystems
subsequent
there
lence
"effect",
that
that
further
are
instruments
measured
notion,
f.
there
assumption,
and
It was
of
microsystem,
other
"state"
ensemble
each
such
Like
fundamental
[5].
Assume,
(Actually
another
operationally.
Haag and Kastler
Assume
as
by the instruof the origi-
apparatus
belongs
defining
in general
f
to a
the new
as
a
certain state
preparing
instrument
the new preparing
instrument
W.
w and w.
14
We
will
with
show,
another
preparing
however, preparing
instrument
the
new
the
initial
that
state W
inequivalent.
instrument
w'
W)
w'
equivalent
depends
state
combining
on
f and
only.
Accordingly,
To
the
equivalent
to w the
show
there
given
or,
effect to w
apparatus
leads
to a new
in
other
words:
equivalence
class
of w
this,
exists
assume
at least
w
and
that (i.e.,
w'
one effect
f
to
be
apparatus
g, such that
~(g,]) ~ ~(g,]') It
is
the
perfectly
legitimate,
instruments
give
the
gered),
result
however,
f and
g as another
"yes"
("no")
irrespective
to
consider
apparatus
if its "g part"
of the response
in terms of this effect apparatus
of its
the
combination
of
is defined
to
h, which
is triggered "f part".
(not trig-
If reformulated
h, the above inequality
becomes
~(h,w) ~ ~(h,w') , which is a contradiction
since w and w' are equivalent.
Whereas
thus
the new state W depends
ment
w
only
through
for
the
effect
f
within
tuses
its
apparatus the
on the initial
equivalence f does
same
class,
not
hold.
equivalence
the
preparing
analogous
Indeed,
class
statement
different
commonly
yield
instru-
appara-
different
states W, as will be shown later.
An ing
apparatus a given
other
which
ensemble
ensemble
depending Keeping
f applied
on the
-
of N
again
f and
W,
apparatus
it is applied,
tain mapping
in the way
described,
systems
in
some
consisting
of
N
is said f
just
fixed
to perform and
a non-selective
thereby
initial
systems
-
state W into in
a new
a non-selective
varying
the
operation
initial thus
transforman-
state
operation. state
generates
W
to
a cer-
15
: W + W : ~W
(2.1)
of the set K(H) of normalized
density operators
The
all
mapping
~
thus
the given effect physical
describes
apparatus
procedure
possible
(states) into itself.
state
changes
f. As the mathematical
described
above,
this
induced
counterpart
mapping
~
will
by
of the also
be
called here a "non-selective operation".
By using the same effect apparatus
f in a slightly different way, one
can also perform a selective operation,
defined operationally
as fol-
lows. When applied to an ensemble of N >> 1 systems in a state W, the apparatus
f will
after
their
which
have
systems, stems.
be
triggered
interaction triggered
with
f, while
in N+ = ~(f,w)N the
apparatus
disregarding
we arrive at another ensemble, (With
~(f,w) = 0,
N,
an
N+
can
exceptional
also
be
case
to
cases.
f,
the
By
N+
selecting,
microsystems
the remaining
N_ = N - N+
now consisting of N+ microsy-
made be
arbitrarily
considered
large
unless
separately.)
This
^
ensemble
is
also
both applied the
in
a well-defined
instruments,
instruments
w and
new
state W,
depending
again
on
w and f, since as above the combination of
f - now
applied
in a somewhat
different
way,
however - may be considered as a new preparing instrument ~ belonging ^
to a well-defined equivalence class W. ^
As
before,
only,
rather
consider systems cases
the new
state W depends
on
the equivalence
class W of w
than on the particular w chosen from it. To
the application prepared
by
the
of an arbitrary procedure
~.
show this,
effect apparatus g to the N+
Denoting
by N++
the number
of
in which the apparatus g is triggered by this ensemble, we have
by definition ~(g,~) = N++/N+ .
Consider again the combination of the effect apparatuses ^
(2.2)
f and g as a ^
new effect apparatus h, but define now the effect to be measured by h
16
to
occur
if
apparatus
and
are
only
if
triggered.
both
the
Since
f
thus
and
g
used
part
in
a
of
the
different
combined way,
the
^
combination different
of from
discussion riment
of
f and
f
and
g
now
becomes
the apparatus
of combined the
kind
g apparatus
an
effect
h considered
yes-no
above.
measurements
considered,
the
see
either
(For
§6.)
successive
may be interpreted
apparatus
h which
is
a more detailed
In a single
triggering
expe-
of both
the
as a triggering
of g by a
~, or else as a triggering
of h by a
^
system
prepared
by
the procedure
system
prepared
by
the
sponds
to Eq.
instrument
(2.2), whereas
w.
The
the second
first
interpretation
one yields
N++ = ~(h,w)N
.
N+ = ~(f,w)N
,
corre-
the relation
Since
we may rewrite
(2.2)
in the form
~(g,~)
The
right
arbitrary also
hand w'
side
in
of
the
unchanged,
for
= ~(h,w)/~(f,w)
this
same
is
unchanged
equivalence
arbitrary
.
g;
if
class
i.e.,
(2.3)
w
W.
this
is
replaced
Therefore
replacement
by
an
~(g,~)
is
leads
to the
^
same state W.
If ~(f,w)
= 0,
we
use
and
f
of
above
w
does
not
have
N+
according
really
lead
= ~(f,w)N to
the
= 0 as well; selection
to a preparation
i.e.,
the combined
prescription
procedure
described
~. Accordingly,
^
a
"new
state"
W
resulting
and need not - be defined
In
contrast
plied
from
"selective
operation"
can
not
-
in this case.
to a non-selective
to N microsystems
the
operation,
in a state
W,
a
does
selective not
operation,
always
ap-
lead
to N but
The
fraction
^
in
general
N+/N
only
to
N+ < N
is called - quite
systems
legitimately,
in
the
new
of course
state
W.
- the transition
proba-
17 ^
bility
of
exists,
the
and
state
coincides,
the probability it
depends
already
W
the "transition
probability
W,
change
stated
effect for
(If
N+
probability"
= ~(F,W)
the
above
^
W.
by definition
~(f,w) on
+
=
0,
no
is zero.)
final
As this
of the selection
state
transition
procedure,
with
of the effect F in the initial
apparatus
non-selective
f up
to
equivalence
operations,
W
state
only.
however,
As
a corre-
^
sponding
statement
for
the
final
state
W
itself
will
turn
out
to be
wrong.
Mathematically,
a
selective
operation
could
be
identified
with
the
^
mapping before ges,
W ÷ W of
initial
into
for a non-selective
however.
First,
such
final
states
induced
by
it,
as we did
operation.
This would have some disadvanta-
a mapping
would
not
be
defined
on
all
of
^
K(H),
since
mapping
W
does
not
would not specify
re would
have
selective "selective
to
be
operation
exist
operation"
~(f,w)
the transition
given a
if
separately.
slightly in the
= 0.
probabilities, We
different
following
= ~(F,W)
therefore mapping
- which
Second, which
therefo-
associate
~ -
also
is defined,
this
with
a
called
a
for arbi-
trary initial states W ~ K(H), by
= f~(F,W)W
~w
if ~(F,W) ¢ 0 (2.4)
I
0
if ~(F,W) = 0 ^
in
terms
of
the
final
states
W
and
the
transition
probabilities
^
~(F,W).
Since trW = 1 by definition,
~(F,W)
both the transition
probability
= tr(~W)
(2.5)
and - if it exists - the final state ^
W = ~W/tr(~W)
can
then
mapping ~.
be
calculated
for
an
arbitrary
(2.6)
initial
state
W
from
the
18
According K(H)
to its definition
into itself,
thus Hermitean) rators)
with
but rather
~ does not map
the set of states
into the set B(H)~ of non-negative
trace class operators
(or "unnormalized"
tr(~W) ~ I. In view of the additional
tion provided by Eq. a
(2.4),
disadvantage.
(2.5), however,
The
(2.4) is, however,
most
(and
density ope-
physical
informa-
this is an advantage rather than
important
motivation
that ~ is convex-linear,
for
the
definition
i.e., satisfying
~(%W 1 + (I-I)W 2) = I~W 1 + (I-I)~W 2
(2.7)
for arbitrary WI, W 2 e K(H) and all real ~ with 0 < % ~ i.
For
the
proof
of
(2.7),
ties tr(~W I) and tr(~W2)
assume
first
that
the transition
are both non-vanishing.
probabili-
As already discussed
in §I, an ensemble of N systems in the mixed state
W = %W 1 + (l-%)W 2
may
be prepared
the
state
W 2.
(Since
WI,
by applying and
%N = N 1 times an instrument w I preparing
(I-%)N = N 2 times
(2.7)
is
trivially
another
satisfied
instrument
for
~ = 0
w 2 preparing
or
i,
we
also
assume 0 < ~ < i.) Applying the selective operation ~ in this particular
case
then amounts
w2 bY ~I and ~2' apparatus
f
to replacing
respectively,
and
the
the preparing
by adding
above
selection
instruments
w I and
to each of them the effect prescription.
The
state
^
W = ~W/tr(~W)
of
the
ensemble
after
the
operation
~
is
thus
some
mixture, ^^ 1 + (I- f)w 2 ' W^ = %W ^
of
the
states
^
W 1 = ~Wl/tr(~Wl)
and
W 2 = ~W2/tr(~W2)
prepared
by
~)I
and W2' respectively.
The
instrument
surviving
wI
prepares
N1
systems,
NI+ = tr(~Wl)N 1
of
the selection procedure performed with the apparatus
them f, and
19
thus
leaving
the
N2+-- tr(~W2)N 2 NI+ + N2+
preparing
systems.
systems;
instrument
~I"
final
ensemble
The
so the
transition
Similarly, thus
probabilty
of
~2
prepares
contains the state
N+ = change
^
W+Wis tr(~W) = N+/N .
Moreover,
the weight factors in the m i x e d s t a t e
W become
= NI+ = tr(#Wi)Nl = tr(#Wl) N+
tr(~W)N
tr(~W)
and 1
~
tr(~W2)
-
(i
:
-
~)
.
tr(~W) This, finally,
implies ^ ^
~W = tr(~W)W = tr(#W)(%W 1 + (I-~)W 2) ^
= %tr(~Wl)W 1 + (l-%)tr(#W2)W 2 = %~W I +
(I-%)~W 2
,
i.e., Eq. (2.7).
The same result is also obtained in the exceptional cases
tr(#W I) # 0 ,
tr(#W 2) = 0
and tr(~Wl) = tr(~W 2) = 0 .
As
#W 2 > 0 by definition,
case
we
therefore
= (NI/N)(NI+/NI)
get
tr(~W 2) = 0
N2+ = 0,
and
implies thus
~W 2 = 0.
~ = 1
and
= ltr(~Wl) , which implies (2.7):
~W = tr(#W)W 1 : %~W 1 = %#W 1 + (I-I)~W 2 •
In the
first
tr(~W) = NI+/N
20
In the
second
tr(~W)
a
continuing
of
finds
some version
follows: systems
If
we
In
a
effect
shall
in the
E
are,
the
the general
used
is
measured
N+ = tr(EW)N
after
the
one
formula",
here
proper-
discussion
mechanics
reduction
language E
the mathematical
quantum
packet
effect
then
of
trivially.
of
illustrate
textbooks
decision W,
(2.7) holds
investigation
of the "wave
which
in state
the
our
~,
example.
postulate",
red
with
the mapping
specific
tion
~W 1 = ~W 2 = 0 and - since N+ = 0, and thus
= 0 - also ~W = O, so that
Before ties
case we have
by
usually
or "projec-
can be formulated in
systems
an
ensemble
which
measurement,
in
have
the
as
of
N
trigge-
new
state
^
W = EWE/tr(EW).
Since
in
(2.4)
the
leads
case
considered
the
tansition
probability
The corresponding
non-selective
.
(2.8)
operation
is
~W = EWE + (I-E)W(I-E)
ing
tr(EW),
to ~W = EWE
Namely,
is
the
projection
the non-occurrence
N_ = tr((I-E)W)N
postulate of E,
systems
applied
implies
which
have
to
that, not
.
(2.9)
the
effect
after
I-E,
describ-
the measurement,
triggered
the effect
the
E are in
^
the
state
W' = (I-E)W(I-E)/tr((I-E)W).
Therefore,
if
no
selection
is
^
made,
the
state W = ~W after
the E measurement
is a mixture
of W and
^
W'
with
the weight
factors
tr(EW)
and
tr((l-E)W),
respectively,
and
is thus given by (2.9).
It
is
commonly
admitted
that,
in
practice,
there
are
also
E measure-
^
ments
which
(2.9).
do not
For this
lead
reason,
to the final a measurement
late
is sometimes
called
that
such
measurements"
"ideal
an
"ideal are
states
W or W given
satisfying
measurement". only
a
very
by (2.8) and
the projection We shall
postu-
indeed
particular
show
type
of
21
operations,
and
that
actual
not
very
likely
to be of
are
not
decision
effects
anyway. mind
Nevertheless,
as simple
this
type.
Eqs.
the
examples
measurements
decision
Besides
(2.8)
examples
of
and
to effects
(2.9) would
(2.8)
illustrating
this,
and
E
are
F which
not be applicable
(2.9)
the general
effects
should
be kept
discussion,
in
to which
we now return.
By virtue
of
(2.7),
the mapping
unique
way
to a mapping
trace
class
B(H) 1 into
I.e.,
the extended
-
for
¢ : K(H)
÷ B(H)~
simplicity
also
itself,
mapping
which
is
can be extended
denoted
by ¢ - of
complex-linear
and
¢ : B(H) 1 ÷ B(H) I satisfies
for all T, S g B(H) I and all complex
CT ~ 0
(2.10)
(2.18)
and
(2.11)
(2.10)
- or else
below - ~ is also real,
numbers
if
by
a and b; and
T ~ 0 .
its
(2.11)
explicit
construction,
cf.
see
this,
T+ > 0. by may
By
(2.10), be
note
that
(2.11), then,
written
the
any
(2.12)
Hermitean
operators
~T = ~T+ - ~T_ as
Eq.
i.e.,
CT * = (¢T)*
To
the
positive.
¢(aT + bS) = aCT + b¢S
By
in a
~T+ is
T 1 + iT 2 with
T
is
are also
of
the
> 0, and
form
T 1 and
with
thus Hermitean,
Hermitean.
Hermitean
T+ - T
and
A non-Hermitean T2,
so
that,
T
again
by (2.10),
~T* = ¢(T 1 - iT 2) = CT 1 - i~T 2 = (~T 1 + i¢T2)*
Although
the
standard,
we
procedure shall
done in three steps.
of
sketch
extending it here
for
~
from
K(H)
the reader's
to
= (¢T)*
B(H) I
.
is
quite
convenience.
It is
22
In a first of
the
step,
cone
the original
B(H)I
of
mapping
"unnormalized"
which is also convex-linear
~ is extended density
to a mapping
operators
into
(or rather "positive-linear");
~+
itself,
i.e.,
~+(T + T') = #+T + ~+T'
(2.13)
~+aT = a~+T
(2.14)
and
for
all
T, T' s B(H)~
and a l l
numbers
a > 0.
To
achieve
this,
set
~+T = 0 if T = 0, and
(2.15)
~+T = trT.~(T/trT)
if
T ~ 0,
the
last
if 0 # T e B(H)~. T, T' e B(H)~
expression
From this,
(and
neither
being
(2.14) of
them
well-defined follows equal
since
trivially. to
zero,
T/trT g K(H)
Moreover, since
with
otherwise
(2.13) holds trivially), we get T + T' V d~.
trT
= tr(T+T')
T --+ tr(T+T') trT
= ~w
with
V, W = T/trT
+
trT'
T '
tr(T+T') trT'
(l-~)w'
and W' = T'/trT' g K(H)
and 0 < X < I, so that,
by
(2.7) and (2.15),
¢+(T + T') = tr(T+T').%V = tr(T+T')(X~W + (I-X)~W')
-- trT.~W + trT'.~W' = ~+T + ~+T' ,
which proves (2.13).
In a second step, of
the
itself.
real
the mapping
space
B(H) h
If T = T* g B(H)I,
of
~+ is extended to a real-linear mapping Hermitean
trace
we have T = T + -
class
operators
into
T_ with T_+ e B(H)I;
take,
23
e.g., not
T± = (ITi ± T)/2 unique,
with
however;
[TI = (T2) I/2.
e.g.,
with
Such d e c o m p o s i t i o n s
arbitrary
T = (l+c)T+ - (T_ + cT+). Nevertheless,
c > 0
we
also
= S+-
S
with
T+, S± ~ B(H)I,
(2.16)
of T is unique. we
have
have
~r defined by
~r T = ~+T+ - ~+T_
with any of these decompositions
are
For,
if T = T ~ -
T+ + S_ = S+ + T
E B(H)I
T_ and
thus, by (2.13),
~+(T+ + S_) = ~+T+ + ~+S_ = ~+(S+ + T_) = ~+S+ + ~+T_
,
or ~+r+ - ++T_
= ~+S+ - ~+S_
There remains to prove real-linearity of ~r' i.e.,
~r(aT) = a~rT ,
~r(T + T') = ~r r + ~r T'
(2.17)
for all T, T' E B(H)~ and all real a. We have aT = S+ - S_ with
S+ = ±aT± ,
the upper
(lower)
signs
+ S_ = ±aT¥ ~ B(H) I ,
being valid
if a > 0 (a < 0), and thus, with
(2.14) and (2.16),
~r(aT) = ~+S+ - ~+S_ = ± a~+r± ~ a~+r¥
= a(~+T+ - ~+T_) = a~rT
.
Moreover, with the decomposition
T + T' = (T+ + T~) " (r_ + Ti)
we obtain from (2.13) and (2.16)
24
Cr(T + T') = ¢+(T+ + T$) - ¢+(T_ + TJ)
= ¢+T+
Finally, defining,
~r
is
with
- ~+T_ + ¢+T$ -
extended
to
the unique
a
mapping
decomposition
¢+T1 ~c
= Cr T + Cr T'
of
B(H) I into
T = T1 +
•
itself
by
iT 2 of T e B(H) I
into Hermitean T I and T2,
~c T = ~rTl + i~rT 2 •
(2.18)
" ' and using (2.17), we then get With T' = T 1' + IT2,
~c(T + T') = ~c(Tl + T{ + i(r 2 + Ti))
= ~r(Tl + r{) + i~r(r 2 + T~)
= ~rrl + i~rr 2 + ~rrl + i~rT ~
= ~c T + ~c T'
.
(2.19)
Moreover, with real ~ and 8,
(~ + iS)T = (~T 1 - ~T2) + i(~T 2 + BT I) , and thus, again by (2.17),
~c((~ + iB)T) = ~r(~Tl - 8T 2) + i~r(~T 2 + 8T I)
= ~rT1
- 8~rT 2 + i ~ r T 2 + iS~rT 1
= (~ + i~)(¢rT 1 + iCrT 2)
= (~ + iS)~cT .
(2.20)
25
Eqs.
(2.19)
and (2.20) are equivalent
to (2.10),
the complex-lineari-
ty of ~c"
If
T = T*,
(2.18)
yields
~c T E ~r T.
~r T = ~+T if T e B(H)~,
and (2.15)
~+'
are
really
of #c (Eq.
(2.11))
~r
and
~c
Likewise,
implies
(2.16)
implies
~+T = ~T if T c K(H).
extensions
of
the
original
Thus
mapping
: K(H) + B(H)~.
Positivity
#+ on B(H)~, done
in
and ~+T > 0 by
(2.10)
identifying
and
follows (2.15).
(2.11)),
an operation
we
with
immediately
since ~c reduces
As announced
shall
omit
the
to
before
(and already
suffix
in ~c'
the extended mapping
thus
~ : B(H) 1 + B(H)I,
from now on.
We
shall
now prove
that
the mapping
~ is continuous
with
respect
to
the trace norm: IJ~T;II < C HTII1 ,
We recall here the definition
with
some properties
[4].)
Denoting
,
ITi = (T'T)
of the trace norm
by B(H)
Sup tr(~W) < I . W~K(H)
the algebra
1/2
,
11-I[I • (Compare
Jtr(XT)j <
llXl; JiTl;
1
for arbitrary X e B(H), T e B(H)I, and IITIII -- Sup Itr(XT)~, IIXll=I Sup itr(XT)I = liTU i=I
[3], Ch.
of all bounded operators
by 11.11 the operator norm, we have
UXI[ =
(2.21)
(1.9),
lITJl1 = trlTl and
C =
Sup itr(XW)i. W~K(H)
1, and
on H and
26
The existence of C =
as required
in (2.21),
tely from the physical lity.
(That tr(~W)
Sup tr(~W) WeK(H)
as well as the statement C ~ I, follow immediainterpretation
is bounded
of tr(~W) as transition probabi-
from above,
teed by the positivity of ~. See, e.g.,
however,
is already guaran-
[3], Ch. 2.)
In order to prove (2.21), we start with the decomposition
T = T+ - T_ = trT+.W+ - trT .W
,
(2.22)
with T+
of an arbitrary
= (iTi
Hermitean
+ T)/2
,
W+
= T+/trT+
T ~ B(H) I. (If,
e.g.,
,
T+ = 0, take W+ arbi-
trary.) Then = T+ + T
ITl
,
;ITII1 = trT+ + trT_ ,
(2.23)
and ~T = t r T + . ~ W + -
trT_.~W_
.
Therefore we get, for all T g B(H)~, ;ICTI;1 =
Sup ttr(X'¢T) i I;XII=I Sup (trT+Itr(X.~W+)l IIXll=1
+ trT_Itr(X.~W_)l)
C(trT+ + trT_) = C llTllI ,
(2.24)
since tr(X.#W+)
Denote
by B(H) h the
~ iiXll llqbW+ll1 = tr(~W+) < C .
set of all bounded
consider a fixed X g B(H) h. Then
Hermitean
operators
on H, and
27
^
x(T) d~. tr(X-¢T) , with T e B(H)~ arbitrary,
defines
a real and real-linear functional
on B(H)~. Moreover, by (2.24),
I x(T)l = Itr(X.~T) l ~ iiXll iI~TI[1 ~ C liXJi flTll1 ; ^
i.e., x(T) is continuous with respect to the trace norm. Therefore it is of the form x(T) = tr ( x T) ^ with a unique operator ^
X d~. ~*X e B(H) h . that the real Banach space B(H) h
Here we have made use of the fact
(with the norm II.I[) is the dual of the real Banach space B(H)~ (with the
norm
II-;Ii), in the following
functional
x(T)
on
B(H)~
which
sense: is
Every
continuous
(real-)linear with
respect
(real) to
the
trace norm is of the form tr(XT) with a unique X e B(H)h; vice versa, tr(XT) with an arbitrary X e B(H) h defines such a functional x(T) on B(H)~, and the norm llx;1 =
Sup Ix(T)l IITIIi=i
of this functional coincides with liXl[ (cf. [3], Ch. I). ^
Varying X in the above construction,
we obtain a mapping ~* : X ÷ X
of B(H) h into itself. By definition,
tr(X-~T) = tr(~*X-T)
(2.25)
for all X e B(H) h, T e B(H)~. In this sense, ~* is the adjoint of the mapping
~ of B(H)~
into
itself.
(Strictly
here with the restriction ~r of ~ to B(H)~; called ~ ) .
speaking,
we are dealing
thus ~* would better be
By (2.25), the mapping ~* is real-linear, and can thus be
28
extended itself,
in
a
unique
which
for
way
to a
complex-linear
simplicity
is
also
mapping
denoted
of B(H)
by
#*:
into
For
any
B(H)h, def'ine
X = X I + iX2 e B(H), with (unique) Xl, 2 e
~*X = ~*X 1 + i~*X 2 .
Complex-linearity
then
follows
as
above
for
the analogous
(2.25)
becomes
extension
~c of ~r" With
this
X e B(H)
extension
of ~*,
then,
and T e B(H) 1 - or,
in other words:
ping #* of the complex Banach plex-linear complex
mapping
Banach
space B(H)
valid for arbitrary
the complex-linear
is the adjoint
map-
of the com-
~ (E ~c ) of the complex Banach space B(H) I. (The
space
B(H)
is
the
dual
of
the
complex
Banach
space
B(H)~ in the same sense as explained above for the real Banach spaces B(H) 1 and (2.25)
B(H) h,
follows
cf.
[3],
Ch,
I, or
easily by inserting
T = T1 +
iT 2 e B(H) 1 into
resulting
expressions;
they
The
extended
validity
of
arbitrary X = X I + iX 2 e B(H) and
tr(X.~T) are
[4].)
and tr(#*X.T),
equal
since,
as
and comparing is
already
the
known,
tr(Xi.~Tj) = tr(~*xi, rj) for i,j = 1,2.
The mapping #* is continuous with respect to the operator norm: II~*XJJ = Sup Itr(~*X'W) i = Sup Itr(X.~W) J WEK(H) W Sup IJXII II#WII1 = IJXil Sup tr(~W) -- C IIXll . W W Thus, finally, I[~TI[1 = Sup Itr(X'¢T) J = Sup Itr(#*X'T) I lJXI[ =1
Sup ll~*Xll ilTllI < liXll =I
which completes the proof of (2.21).
[[XI[ =1
C IlTII1 ,
(2.26)
29
Like ~, its adjoint
~* is positive:
¢*X ~ 0
and - therefore,
if
or by the explicit
X ) 0 ,
(2.27)
construction
¢*X* = (¢*X)*
above - real:
.
For if X ~ 0, we have tr(~*X.W) for
all
W e K(H),
with an arbitrary
The mapping F
which
belonging
to the operation
arbitrary
~*X > O.
an explicit
(Take,
e.g.,
representation
~. By (2.5)
= tr(~W)
W ~ K(H).
> 0 W = J f >< f J
f g H.)
provides
~(F,W)
for
implies
unit vector
~* also
= tr(X.~W)
and (2.25),
= tr(l.~W)
We may
thus
of the effect
we have
= tr(~*l-W)
represent
the
given
effect
F by
the operator F = ~*i
(for which,
therefore,
ty function
~ takes
we use the same
already
have
F
mentioned
= F*
>
0,
(Eq.
and
F indeed
ced in §i (Eq.
(1.15)).
effect
apparatus
f,
which so
= tr(FW)
(1.16)).
tr(~W)
F ~ i, so that
Every
symbol),
so that the probabili-
the form
~(F,W)
as
(2.28)
=
belongs
tr(FW)
this
~
~* is real
and
I for
W ~ K(H)
to the set L(H)
can be measured
that
Since
,
apparatus
all
of operators
"non-destructively" can
be
used
positive,
implies introdu-
by some
to perform
we
effect
a selec-
30
tire
operation
~,
is
thus
described
given by (2.28). As mentioned assume tion
that is
effect
such
not can
consider ~F(W)
even be
probability
f exist
represented
p(F,W)
by
an
an
for
operator
takes
the
arbitrary
of
Eq.
(I.I0)
in
terms
but
of
operator
F c L(H)
This assump-
that
an arbitrary
such
tr(FW).
fixed
effect
To F
that
show
the this,
as a function
W. The physical mixing
to
F.
F a L(H),
form
state
class
proving
on the set K(H) of density operators
tation
unique
it would not be unreasonable
however,
~(F,W)
with
before,
a
in every equivalence
necessary,
function
by
implies
interpre-
that
PF
is
convex-linear,
~F(%WI + (I-X)W 2) = %PF(WI) + (I-%)PF(W 2) •
Therefore may
the
extension
be applied,
(first)
procedure
in exactly
to a real-linear
described
the same way,
functional
above
for
the mapping
to the function PF'
on B(H)~ - also denoted
leading by PF -
which is positive: PF(T) > 0
(The
further
extension
on B(H) 1 is not needed
if
T > 0 .
of ~F to a complex-linear here.)
Since PF(W)
(2.29)
positive
functional
< I for all W E K(H),
we
obtain, by using (2.22) and (2.23) for an arbitrary T E B(H)~, IPF(T)I = Itrr+.~F(W+) - trT_.~F(W _)I
i.e.,
the functional
<
trT+'PF(W +) + trr_-PF(W)
<
trT+ + trT_ = IITIII ;
PF on B(H)~
is trace norm continuous.
Therefore
it is indeed of the form ~F(T) = tr(FT)
with a unique F ~ B(H) h. Finally,
(2.29) implies F ~ 0, whereas F < 1
follows because tr(FW) = ~F(W) = p(F,W) < I for all W.
31
Regardless way,
of
the
effect
operators
or in the form F = ~*i as described
conditions F
whether
e
for
L(H);
in
operators. than,
these
particular,
This
e.g.,
of effect
operators
already
the subset
operators
which
they
indicates
that
With
G e L(H),
(2.27)
~*(I-G)
~ 0,
i.e.,
effects
into
itself.
tation.
Imagine
state to Let
W,
the
one
the
F
We by
N+ = tr(FW)N
and then
the
G
has
with
ask:
Therefore,
the
as
~*
a
projection
L(H)
-
rather set
supporting
by
often
maps
simple
the
effect are
Now,
set
physical
of
interprein some
~,
corresponding
effect
G afterwards.
apparatuses f
f
triggered
is
are
in of
and
f and
both
number
also
L(H)
of N systems
other
afterwards required
I - G ~ 0,
operation
some
microsystem?
which
be
set
any
requirement
arguments
and,
selective
how
finds
- is the "natural"
an ensemble
be measured
same
the apparatus
~*G
the
to
whole
More
Thus
and measures
systems,
W = ~W/tr(FW). trigger
performs
the
this
later.
~*G > 0
effect
starting
F = ~*i,
respectively. successively
implies
The
that,
effects
the
in
one never
shown
operators
~*G ~ ~*i = F ~ I.
first
effect
be
F in quantum mechanics.
this point of view will be presented
obtained
go beyond
not
of projection
are
before,
would
can
F
g
the
triggered first new
systems
g,
which
by
state also
g is ^
N++ = tr(GW)N+ = tr(G. ~W)N = tr(~*G.W)N
Thus f and g occur successively
~(f,g;W)
The
effect
composite are
effect
triggered
describing H = ~*G. of
apparatuses
the
the effect
= N++/N = tr(~*G.W)
Then,
corresponding the effect
apparatuses
be
h, defined
successively.
Therefore
in the state W with probability
f and g may
apparatus
.
effect
.
considered
(2.30)
as
to be triggered
according (the
the
class
the successive
shall
see,
of a new,
if both f and g
(2.30),
equivalence
~*G describes
f and g. As we
to
parts
operator of
h)
is
triggering
different
opera-
32
tions
~ may
effect
~*G
depends
correspond is
not
to the same effect
fixed
by
on the particular
the
effects
operation
F, which
implies
that the
F
alone,
but
and
G
~ applied - or,
also
in other words,
on the particular effect apparatus f used to measure the effect F.
The
adjoint
itself
is
#*
of
normal;
Ch. 2).
every
positive
i.e.,
it
Consider
Xn+ 1 - X
> O)
n
of
an
has
linear the
increasing
Hermitean
to a bounded
by
ultraweak
called
of
the
e.g.,
normal
of
B(H) 1
into
(cf.
[3],
following
property
sequence
(Xn+ 1 > Xn;
operators
~ Y = Y* ¢ B(H) n weak operator topology
T ¢ B(H) 1 (cf.,
#
i.e.,
e B(H), n = 1,2,..., with n for all n. Such a sequence converges in the ultra-
X
definition
mapping
if,
[3],
for
Ch.
every
X
Hermitean
topology,
operator
X ~ Y;
tr(XnT) ~ tr(XT)
i.e.,
for
all
1). A mapping ~ of B(H) into itself is such
sequence
Xn,
~X n
also
converges
ultraweakly to ~X.
To
prove
implies
normality
of
~*,
note
that
~*Xn+ 1 > ~*X n and ~*X n ~ ~*Y;
positivity therefore
of
#*
(Eq. (2.27))
#*X n has an ultraweak
limit X. This implies, for all T e B(H)I, ^
n+
tr(~*Xn'T) whereas,
,
on the other hand, tr(~*Xn'T) = tr(Xn'~T)
since
tr(XT)
X
+ X n
n
ultraweakly.
Thus
^
n÷ tr(X.~T) -- tr(~*X.T) tr(XT)
= tr(~0*X.T)
for
, all
T,
which
indeed implies X = ~*X.
For
physical
property,
reasons,
called
the mappings
complete
positivity
~ and ~* must have and being somewhat
still another stronger
than
space Hn,
and
"ordinary" positivity as expressed by (2.11) and (2.27).
Consider the
a natural
tensor
product
number
n,
H ~ H
n
of
an n-dimensional the
state
Hilbert
space
H,
of
the
quantum
33
mechanical
system
considered,
with
H . Represent n
vectors
g e H
n
by
column vectors, c1
g =
(ci), c
and operators Y on H
n by
n
n x n matrices,
~ii" " " ~in) y =
•
(aij) anl " " with
complex
numbers
c i and
,
;nn-
aij , as
usual•
Then
vectors
may be represented by column vectors with "components"
,
f = ..[fi ] --
~ e H 8 Hn
in H,
f.1 e H
such that, e.g., (~,$) = ~ (fi,gi) i while
operators
X c B(H ~ H n)
become
,
n x n
matrices
with
operator
valued "matrix elements",
= (Xij)
,
Xij c B(H)
,
with almost obvious calculation rules like, e.g., X f =
(~.
Xijfj)
J In particular,
product vectors
f @ g with f e H and g = (ci) e H n are
represented by f ~ g =(clf)
while
product
take the form
operators
X ~ Y
with
,
X e B(H)
and
Y = (aij) e B(H n)
34
X @ Y = (aijX) .
An operator T on H @ H -
belongs to the trace class, n
! = (Tij) ~ B(H @ Hn) I , if and only if Tij e B(H) 1
i.e.,
if
all
"matrix
for all i,j = 1...n ;
elements"
T.. ij are trace class operators. Any linear combination of (at most n 2) operators
such T is thus a finite
of the form T @ S with T e B(H) 1 and S ~ B(Hn) 1 E B(Hn).
Its trace is
given by trT = ~ trTii . i Now
consider
itself,
an
arbitrary
and define
(complex-)linear
a mapping
(2.31)
mapping
~n of B(H @ Hn) 1 into
~ of
B(H) 1 into
itself - which
is
also linear - by
~n: Then
the
original
T = (Tij)
mapping
+
~nT = (~Tij)
~ is called n-positive
.
(2.32)
if ~n is positive;
i.e., if ~n T ~ 0
and
~ is called
Similarly,
completely
a linear mapping
for
_T ~ 0 ,
positiv e if ~ of B(H)
it is n-positive
for all n.
into itself yields linear map-
pings n : _X = (Xij)
of B(H ~ H n) into itself, n-positive
if ~
÷
~n-X = (~Xij)
(2.33)
for any natural number n. Again ~ is called
is positive, n
it is n-positive for all n [6].
and ~ is called completely
positive
if
35
If
~ : B(H) + B(H)
is
m < n. To show this,
n-positive,
it
is
also
m-positive
for
all
consider operators X e B(H 8 Hn) of the particu-
lar form
with
arbitrary
m x m operator
B(H 8 Hm) , and
zeroes
matrices
everywhere
Y,
else.
representing
Then
X
>
0
operators
on
and
if
if
only
Y > 0; moreover, we have
nThus Y > 0
implies
--
010
X > O,
so that,
thus ~ Y > O; involving
i.e.,
with
~ X > O, and
operator
n -
~ is
also
matrices
m-positive.
with
plies t o m a p p i n g s ~ o f B(H) 1 i n t o
Since
if ~ is n-posltive,
--
"ordinary" n > 1 and,
positivity more
so,
~n a n a l o g o u s
"matrix
elements"
argument -
now
from B(H) I - ap-
itself.
is the same as l-positivity, complete
ments. A physically relevant example
positivity,
are
n-positivity
stronger
require-
of a positive but not completely
positive mapping will be discussed in §3.
A mapping ~ : B(H) 1 + B(H) 1 is n-positive mapping ~* that
: B(H) + B(H) is n-positive.
the mapping
(~*)n defined
by
if and only if the adjoint
We show this by first proving
(2.33)
- with
~* for ~ -
is
the
adjoint of the mapping ~n defined by (2.32); i.e.,
(~*)n = (~n)* "
By definition of the adjoint (see Eq. (2.25)),
tr(X.~nT) = tr((~*)nX. ~)
(2.34)
(2.34) means
(2.35)
36
for
arbitrary
since,
X = (Xij) e B(H @ H n)
according
to
"matrix
and T = (Tij) e B(H @ Hn) I.
calculus"
and
Eqs.
(2.32)
and
But
(2.33),
X-~n ~ and (~*)nX. ~ have the matrix representations
X-~nT = (~ Xik'~Tkj) ,
(~*)nX.T = ([ ~*Xik'Tkj) k
k
,
Eqs. (2.31) and (2.25) indeed lead to (2.35):
tr(X.#nT)
=
[ tr(Xik'~Tki ) ik
=
[ tr(~*Xik'Tki)
= tr((~*)nX.T)
•
ik According to (2.34), we may simply write ~
for (~*)n or (~n)*.
Now let ~n be positive. Then #n~ > 0 for all W e K(H 8 Hn) ; therefore, with (2.35), X > 0 implies
tr(~X.W)
for all W, which Vice
versa,
= tr(X.~n ~) > 0
in turn implies
let #~
be
positive.
¢~X > 0, i.e., Then ~ W
positivlty
> 0 and
thus,
of ¢~.
again
by
(2.35), tr(_W,~nT) -- tr(~*W.T) > 0
for all W s K(H ~ Hn) if T e B(H @ Hn)l, T > 0. Therefore ~n ~ > 0; i.e., ~n is positive. This proves the above conjecture.
As already mentioned,
there are physical reasons for postulating that
every
or,
operation
~ -
equivalently,
its adjoint
~* - has
to be
completely positive [7]. As is well known (and as will be elaborated here
in some more
considered
detail
in §4),
the Hilbert
as the state space of a composite
space H @ H
system I +
n
can be
II, consi-
sting of the system I considered up to now, with state space H, and
37
another
microsystem
II
with
state
space
H .
(Since
H
n
sional,
system II is usually
called
is
n-dimen-
n
an n-level
system.)
The mappings
cn considered above then acquire a simple physical interpretation.
Assume
that
ensemble
there
is
no
of N composite
teracting
systems
I
interaction systems
and
II.
between
systems
then consists
Its
state
I
and
of N pairs
is described
II.
An
of non-in-
by some
density
operator W on H @ H . Now apply to system I of each pair the selectin ve operation described by the mapping ~, while leaving system II unaffected;
or,
in other
with a suitable with
system
words:
effect apparatus
let
system
I of
each pair
interact
f (which is supposed not to interact
II), and select those pairs I + II for which the appara-
tus f is triggered. K(H @ Hn) , which
This procedure
can
be
extended
clearly defines
an operation ~ on
to
linear
a
positive
mapping
of
B(H @ Hn) I into itself.
In
particular,
W g K(H) system,
density
and V ~ K(H n)
operators describe
which may be prepared
W
of
the
uncorrelated
form states
by using separate
II
into
pairs
afterwards.
(See
also
=
W @ V
§4.)
with
of the composite
preparing
w for system I and v for system II, and combining I and
W
instruments
the single systems For
such
states
we
must have ! ( w ~ v) = +w ~ v
by the definition with
V = (vii)
,
of the operation ~. On the other hand,
has
the
matrix
representation
relation, cn(W @ V) = ~W @ V ,
follows from (2.32) for the mapping cn" Thus
cn(W @ V) = ~(W @ V)
(vijW),
since W ~ V an
analogous
38
for
all W E K(H)
and all V s K(Hn).
the last equality tors
W @ S
T g B(H)I, latter,
as
n-positive;
Actually
can be extended,
with and
S e B(Hn) ,
finally
an
preceding
§3,
an
then
all
to
all
to
be
first to all opera-
T @ S
T g B(H ~ Hn) I.
has
(separable)
with
Thus
positive.
stronger
that
this
could
be
Hilbert
rather
to postulate
somewhat
however,
argument
"ordinary",
lead us
to be
by linearity,
arbitrary
~n E i;
Therefore
but
the
~
must
be
well
with
an
of Hn,
as
and n was arbitrary.
infinite-dimensional
would
to
operation,
the
describing
Since both ~n and i are linear,
applied
as
space H in place
than
an
n-level
system
II.
This
a positivity property for ~ which appears
than complete apparently
positivity.
stronger
We
property
shall prove is already
in im-
plied by complete positivity.
The effect
apparatus
f used for the operation
perform
another
selective
Namely,
instead
of selecting,
systems
which
the
systems
have
which
operation
triggered have
not
after
~',
~ may also be used to
called
complementary
to
~.
their introduction with f, those
the apparatus triggered
f.
f, we may as well
By
interchanging
select
"yes"
and
"no" in the verbal interpretation of the response of the effect aparatus f, we get another effect apparatus not
triggered,
and vice
versa.
The
f', which is triggered if f is
corresponding
effect
operators
F
and F' are related by
F + F' = 1 ,
since
the occurence
of either F or F'
(2.36)
is certain
by definition,
and
therefore
tr(FW) + tr(F'W) = tr((F + F')W) = 1 = tr(IW)
for all
W e K(H).
F' = 1 - F
(F e L(H)
describing,
in
clearly
this
sense,
implies the
F' e L(H).)
non-occuurrence
The
effect
of F,
is
39
also called complementary
to F - or, briefly,
"not F". By definition,
and F are also complementary to ~' and F', respectively:
&')'
complementarity
=
~
,
(F')'
is thus a symmetric
=
F
;
relation.
Since
the operation
~'
is performed by selecting according to the occurence of F', two operations
~ and ~'
complementary
complementarity
(2.36)
of
to each other are characterized the
corresponding
effects
by the
F = ~*i
and
F' = ~'*i or, equivalently, by
tr(~W) + tr(~'W) = 1
Whereas, F',
by
there
(2.36),
are
for all W e K(H)
an effect F has exactly
in general
many
to a given operation #, since
different
.
(2.37)
one complementary
operations
(2.37) with given
#'
effect
complementary
~ does not uniquely
determine ~'. Both ~ and ~' thus depend on the particular apparatus f used to measure the effect F, rather than on F only.
Finally,
as already
mentioned
at
the beginning
of
this
chapter,
the
effect apparatus f may also be used to perform a non-selective operation.
In this case,
apparatus joint)
f is made
subensembles
no selection with respect - or, of
equivalently,
systems
in
an
sting
initial again
state
of N
W
is
of which
by
the
(by definition complementary
dis-
opera-
Thereby an ensemble of N systems
transformed
systems,
the two
produced
tions ~ and ~' are mixed afterwards.
to the response of the
into
another
N+ = tr(~W)-N
ensemble, are
consi-
in the
state
^
W = #W/tr(~W),
whereas
the
remaining
N
D
= tr(~'W).N
systems
are
in
^
the (Eqs.
state
W' = ~'W/tr(~'W),
(2.5),
according
to
the
physical
meaning
of
(2.6)) and ~'. The final state W resulting from the non^
selective operation is thus a mixture of the states W and W' with the weights N+/N = tr(~W) and N_/N = tr(~'W), respectively; ^
^
= tr(~W)W + tr(#'W)W' = #W + ~'W .
i.e.,
40
Therefore
the mapping
~
: K(H)
+ K(H)
describing
the non-selective
operation (Eq. (2.1)) is simply given by
$=~+~'
(Since
~ and
~'
are
:w
positive,
÷
we have
trW = i. Thus ~ indeed maps K(H) also
be
extended
to a mapping
~=~w+~'w.
W
(2.38)
> 0, while
into itself.)
of
B(H) 1 into
Like
(2.37)
implies
# and #', ~ may
itself.
As
is obvious
from (2.38), this extension is given by
~--~+~'
: T
+
~=~T+~'T
(2.39)
for arbitrary T e B(H) I.
Since
the adjoint ~*
of $ is ~* + ~'*,
the "effect"
corresponding to
= F + F' = 1 .
(2.40)
the nonselective operation ~ is
= $*I = ¢*i + ~ " i
This is also obvious from (and equivalent to)
tr(FW) E tr(~W) = 1
for all W e K(H)
,
which expresses the fact that the transition probability is identically one,
and is thus
"effect"
F = 1
always
gives
effect
thus
selective
describes
the
result
simply
operations
lar "selective"
characteristic
counts
the
"yes".
of a non-selective
trivial An
"yes-no
effect
the systems
measurement"
apparatus
The
which
~ measuring
in any given ensemble.
can thus be considered
ones - the "selection"
operation.
this
As non-
formally as very particu-
being made with respect to the
trivial effect F = I - they need no separate mathematical treatment.
These considerations are illustrated by the example of "ideal measurements"
(Eqs.
(2.8) and (2.9)).
In this case,
mentary to ~ is given by ~'W = E'WE'.
the operation ~' comple-
41
Summarizing operation
the main
has
tion in
the
For
by
at
more
that
on H
least
knows
~
-
position practice.
that
of
or
an
In
a yes-no
a
measurements
by
few -
a
that
measurement "suitable"
such
an
cases
mapping
we
~ with this
only
these
assump-
remind
projection
could
apparatus,
be
the
even
would
opera-
performed, if
corresponding,
apparatus
assume
in quantum mechanics.
every
which
selec-
we shall
of defending
arguments,
assumes
satisfies
of any additional
every
Instead
positive
which
operations,
versa,
particular
what
itself,
are quite usual
always
principle", in
vice
assumptions
describes
into
realizable"
we may state that every by a completely
the absence
sophisticated
one almost
except
B(H) 1
operation.
less
similar
"in
also,
chapter,
mathematically
"physically
describes
instance,
tor E
for
following
properties
reader
mapping
< I for all W s K(H).
criteria
tion
of this
to be described
complex-linear tr(~W)
content
look
nobody
e.g.,
to
like
in
42
§3 The First Representation Theorem
An arbitrary by
effect
an operator
can be represented,
F on the state
space
as shown in the preceding
H
of the system
shall
now derive a corresponding explicit
trary
operation
, in terms
of operators
considered.
representation on H. This
§2, We
of an arbi-
representation
is
provided by
Theorem 1 (First Representation Theorem): For
an
arbitrary
operation
finite or countably
,,
there
exist
operators
Ak,
k e K
(a
infinite index set) on the state space H, satisfy-
ing Z ~ k~K
~
( 1
for all finite subsets K ( K o '
(3.1)
O
such that,
with arbitrary
T E B(H) 1 and X e B(H),
the mappings
, and
** are given by *T
=
I ~ keK
T ~
**X =
~ ~ kcK
X ~
(3.2)
and
respectively.
In particular,
,
(3.3)
the effect F corresponding
to * is given
by F = *'1 =
If the index set K is infinite, the norm
ordering
I ~ kCK
~
"
(3.4)
(3.1) implies that - independently
of K - the infinite sum in (3.2) converges
topology,
while
the
infinite
sums
in
(3.3)
and
in the trace (3.4) converge
ultraweakly,
thus defining the precise meaning of these equations.
Vice
given
versa,
operators defines
an
~
on
any H,
operation
countably
k
~
K,
,, whose
or even uncountably
satisfying adjoint
condition ** and
are given by (3.3) and (3.4), respectively.-
of
infinite
(3.1),
then
corresponding
set
of
(3.2)
effect F
43
Proof:
We
(3.1)
implies
ly many rem
first
k ¢ K;
uncountably
most
countably
bute
in Eqs.
countable
some
of
the
more
that A k can be different
indices
an
discuss
therefore,
infinite
infinite (3.2)
dense
K
to (3.4).
set
if in the second
since
details.
it
may
part
be
operators
of the theo-
replaced
by
f. ¢ H,
an
at
A k = 0 do not contri-
To prove the above statement,
of vectors
First,
from zero for at most countab-
occurs,
subset,
technical
i = 1,2, ....
consider
Keeping
a
i fixed,
l
and choosing an arbitrary natural number n, (3.1) implies
(fi,A~Akfi)
may
be
true
for
finitely
that
= lIAkfiti2 > I/n
many
indices
k
only.
Therefore
the
subset
K. ~ K defined by 1
l;Akfill # 0
is
at most
countably
kJi K i.
By
implies
A~A k < I, and
bounded,
Assume can
(3.5),
infinite.
Akf i = 0
and thus ~
therefore,
thus
natural
be
thus
The
for
same
all
i
then
if
llAkll < 1, for
in the following, after
a
K = {1,2,...}.
set K = {I,...,N}
k ¢ Ki
(3.5)
is
true
k ~ V i K i. all k.
for But
Therefore
the
union
(3.1)
also
all A k are
= 0 for all k ~ U i Ki, by continuity.
identified,
numbers:
if
that K is countably
suitable
ordering,
with
simpler
case
The somewhat
need not be treated
infinite,
separately,
the
and
set
of
of a finite
since we may formal-
ly enlarge K by setting A k = 0 for k > N.
Consider
the operators Fn
Since n,
and
A~A k > 0, satisfy
these F
< 1
d~.
operators for
all
I A~A k • k
by
n
ultraweakly,
for n + ~, to an operator
(3.1).
and
Therefore
increasing they
with
converge
44
F = ~ A~ A k
n -
which
defines
0 < F ~ i; from
the
i.e.,
B(H),
meaning
F E L(H).
so
of
the
More
that
infinite
sum
generally,
take
0 < X < IIXI;.1.
Since
in
an
(3.4)
-
arbitrary
with X ~ 0
A~ X A k > 0
and
A~(IIXII.1 - X)A k > 0 for all k, the operators ^
Xn
increase
aT.
I A~ X A k k
with n, and satisfy 0 < X
^
n
< IIXiI.F < IIXfl.F for all n. Thus
n
the sequence Xn, n = 1,2,... also converges ultraweakly to some operator ^
X=XA XAk 0 k ^
depends
Since X
linearly on X, and any X E B(H) is a linear combina-
n
tion ly,
^
of at most
four non-negative
operators, (3.3),
thus
giving
a meaning
to Eq.
Moreover,
these
ultraweak
limits
X
for
defining
converges ultraweakn an arbitrary X E B(H).
the
right
hand
sides
of
(3.3) and (3.4) do not depend on the particular ordering of the index set
K.
For
suppose
K
=
{1,2 .... }
to
be
reordered
as,
e.g.,
K = {kl,k2,... }. An arbitrary X > 0 now leads to a sequence of operators
Xn =
~
A~.X Ak.
i
as
above,
converges
{kl,k2,... 1 is a reordering
i
l
ultraweakly of
to
{1,2,...},
an
and
m'
for
each
n,
such
^
Therefore
that
Xn
independence
^
Xm
<
X
Since numbers
^
and
^
Xn
< L'
~ X"
^
also X < X and X ~ X, which
the same holds
X.
there are suitable ^
m
operator
implies
X = X. By linearity,
true for an arbitrary X e B(H), which proves the order
of
the
right
hand
side
of
(3.3),
of
which
(3.4)
is
merely a particular case.
Since Ch.
the I,
or
trace [4]),
class so
B(H) 1 is a two-sided that
XT e B(H) 1
and
ideal
in B(H)
TX e B(H) 1
for
(cf.
[3],
arbitrary
45
T ~ B(H) 1 and X E B(H), we have T
~ n
for all T g B(H)~.
Ak T ~
e B(H)
d~. k n
(Note that, obviously, AkTA ~ > 0 if T ~ 0.) Now we
have m
m
,iTm - Tn"l = I, ~ A k T A~ "I = tr( ~ A k T A~) k=n+l k=n+1 m
-- tr( ~ A~ A k T) -- tr(FmT) - tr(FnT) k=n+l due
to
the
cylic
interchangeability
of
operators
,
under
the
trace,
whereas, on the other hand,
tr(FnT)
÷n
tr(FT)
, ^
by the ultraweak a
Cauchy
convergence
sequence
+ F. Therefore the operators T form n n n respect to the trace norm, so that there
with
^
exists
F
^
a T e B(H) 1 with
^
ilTn - TIll ~ 0, which
gives a meaning
to the
right hand side of Eq. (3.2): ^
T d~. k~ Ak T A~ •
Since trace norm convergence
implies convergence
in the operator norm ^
(uniform)
topology,
and
therefore
also
in
the
ultraweak
topology,
T
^
is also the ultraweak
limit
of the operators T
which,
moreover,
are
non-negative and increasing with n. This yields T > 0 and the indepen^
dence
of T on
class
operators
the order are
linear
right hand side of (3.2) also.
of summation,
trary
T E B(H) 1
(3.4)
shall be understood,
as above.
combinations
exists,
Infinite
of
As
arbitrary
nonnegative
and is order-independent, sums
of
the
type
(3.2),
ones,
trace the
for arbi(3.3)
and
from now on, to represent limits of finite
sums in the appropriate operator topologies.
46
We are now ready to prove the second part of Theorem 1. Assume operators Ak, k e K satisfying ~* given proved;
by
(3.2)
obviously,
and
(3.1)
to be given. Then the mappings ~ and
(3.3) are well-defined
they are also
linear.
and positive,
It remains
as just
to show that ~*
is the adjoint of ~, and that ~ is completely positive.
For arbitrary X e B(H) and T e B(H)I, we have tr(X-T n) -- tr(X. [ % k(n
T ~) ^
= tr( ~ ~ k
X %.T)
= tr(Xn-T)
.
Since tr(XT n) - tr(XT)
=
tr(X(T n- T)) ^
<
^
llXil l i T -
n
TII
i
+
n
0
,
tr(XT n) converges to tr(XT) = tr(X.~T) whereas, since ^
X ultraweakly,
tr(XnT)
^
n
converges
+
X = ¢*X
n
to tr(¢*X.T).
Thus
¢* is indeed
the
adjoint of ~.
To prove
complete
positivity,
consider
an arbitrary
finite-dimensio-
nal or separable Hilbert space H, and define a mapping i of B(H @ ~)I into itself by
±r=
I (Ak S ! ) r (Ak 4)*
(3.6)
keK for arbitrary T e B(H 8 H)I" Since
ksK
(A k @ 4)* (Ak @ i) = ( ~ ~ kgK O
for
any
finite
subset
A k) @ !
~
!
O
K ~ K 0
(with
1 denoting
the unit
operator
on
47 ¸
H 0 H), ~ is a mapping
of the form (3.2) with the operators A k repla-
ced
also
satisfy
! is well-defined,
linear,
by
A k @ 4,
Therefore
which
a
condition
of
and positive.
the
form
(3.1).
If, in particular,
T = T ~ S with T g B(H) 1 and S ~ B(H) I, (3.6) yields ~(r @ S) = I
(A k @ !)(T @ S)(A k @
kEK
~)*
= ( ~ Ak T A~) ~ S = ~T ~ S • keK Taking means
for
H
an
n-dimensional
that i coincides
re, by linearity, introduced ve,
Hilbert
on operators
space
Hn,
the
last
equation
of the form T ~ S - and therefo-
on the whole space B(H 0 Hn) 1 - with the mapping ~n
in §2 to define n-positivity
~ is n-positive,
of ~. Since ~ = ~n is positi-
and thus completely
positive,
since n was arbi-
trary.
Actually
the
above
again
the
form
of
the mappings
which
is
even
also
completely
as operations.
sight
this positivity
~ by postulating
more.
First,
Second,
~ of the form looks
i
is
Therefore
and can indeed
since H may also be
(3.2)
somewhat
since
positive.
in §2 share this property,
mappings
at first
proves
As was already mentioned
ted to include operation
it
physically
infinite-dimensional,
positivity.
(3.2),
~n introduced
be interpreted
property
reasoning
have a positivity
stronger
than
complete
in §2, one might have been temp-
property
the existence
in the very definition of positive
of an
linear mappings
of B(H ~ ~)I' which satisfy ~(W @ W) = ~W @ W for
uncorrelated
states
W ~ W,
for all
finite-
(3.7) or infinite-dimensio-
nal Hilbert spaces H. (The effects F = ~*~ corresponding
to i satisfy
tr(F(W ~ W)) = tr(i(W @ W)) = tr(~W ~ W) = tr(#W) = tr(FW) = tr((F ~ !)(W @ W)) which implies - for details see §4 -
,
48
F ffi F @ 1 .
(3.8)
The same follows from the explicit form (3.6) of ~, which yields
ksK
kcK
in analogy to (3.4).) a
given
infinite-dimensional
B(H ~ ~)I
(as explained
dimensions postulate we
are
that
the
give
of
positivity
for ~, which
the of
then,
thus
property.
complete
positivity
has
also
a
~
is
in §2 for the case of finite
part
just
been
postulate
literature
of
already
proved,
in
to
to
derive
Eq.
implies
the
the
latter,
by mathematicians,
operations
(cf., e.g.,
1 demonstrates
also
contrast
studied
for
Theorem
sufficient
been
Moreover,
already
standard
in the physical
first
as has
positivity
maps
is contained in n B(H ~ Hn) 1 is embedded in
and H), it would also have been sufficient to n for infinite-dimenisonal H only. However, the proof
(3.7)
about
now
and
in more detail
stronger
is
H,
H
of both H
complete
(3.2)
Since any finite-dimensional
[7],
and
related
and state
[3], Ch. 9, [8] and
[9]). The basic ingredient
for the proof of the remaining part of Theorem 1
is Stinespring's C*-algebra
Theorem:
B K B(H)
A
complex-linear
into B(H)
mapping
is completely
~
positive
of
a
concrete
if and only if
it is of the form
with
a
representation
: X s B
+
~
on a
of B
TX = A* ~(X) A
Hilbert
space
H,
and
a bounded
operator A mapping H into H.
Rather or
than proving this here (for proofs,
[3],
Ch.
9), we shall only explain
see the original paper [6]
the basic notions.
A concrete
C*-algebra B is a C*-algebra of operators on a Hilbert space H, i.e., a
subset
of
B(H)
which
is
closed
under
multiplication,
Hermitean
49
conjugation,
and
with
to the uniform
respect
tains
the
unit
C*-algebra.
A
complex
linear
operator
combination
(operator
I.
of operators,
norm)
In particular,
representation
~
of B
on
topology, B(H)
and which
itself
a Hilbert
as well
as
con-
is a concrete
space H
is a com-
plex-linear mapping : X ~ B of
B
into
B(H)
~(X*) = (~(X))*
which and
+
~(X) e B(H)
preserves
~(1) = ~
operator
(the
unit
products
operator
~(J) ~ B(H) of B under n is then also a concrete
We shall, moreover,
also use the following
Lemma
i: Let ~ be a representation
there
are
mutually
sometimes set),
even
which,
dlrekt
sum
operators with H acts
empty
k~K Hk
dimensional
subspaces
the
that w(X),
identical
on a Hilbert
complement
Hk,
All
are
subspaces
when
(The
image
Hk
i.e.,
(a
finite
infinite H
o invariant
restricted
representation;
space H. Then
k e K
uncountably
orthogonal
X e B(H).
H).
C*-algebra).
H k ~ H,
or
subspaces
satisfies
lemmas:
of B(H)
countably
these
all
a way
the
or
with
of
for
in such
restriction
-
together
n(X)
like
orthogonal
on
and
in H
can be
to any
-
index of
the
under
the
identified
such H k E H,
~(X) = X
on
~.
The
w (X) of ~(X) to H satisfies ~ (E) = 0 for all finiteo o o projection operators E. In other words, the orthogonal
decomposition = H ° • ( • Hk), keK of H induces
the reduction
= ~
• ( • Wk ), o
of
the
consists
representation of
representation to
the
on H.)
Hk ~ H
the
zero
w.
The
vector
of the quotient
norm-closed
Uk(X) E X
kCK
two-sided
subrepresentation only. algebra ideal
w
(Technically B(H)/C(H) C(H)
of
is absent o speaking, ~
of B(H) all
with
compact
if is
H
o a
o respect
operators
50
For a proof see, e.g., [i0]. (Compare also [3], Ch. 9.)
Lemma
2: Let H I be a separable Hilbert
space,
and A a bounded linear
operator from H I into a Hilbert space H 2 which need not be separable. Then T ~ B(HI) I implies ATA* e B(H2)I, and
tr2(X.ATA* ) = trl(A*XA.T )
for
all X E B(H2).
for
clarity,
(Here
the traces
by corresponding
in H 1 and H 2 are distinguished,
suffices.
If H I = H2,
Lemma
2 reduces
to well-known facts.)
Proof of Lemma 2: Since A is bounded,
it maps a dense set of vectors
in H 1 into a set of vectors
in H 2 which is dense in the range AH 1 of
A.
thus
As
H I is
separable
separable,
it
follows
that
AH I is contained
subspace H i of H2 ; i.e., A = E2A with
in a
the projection opera-
tor E 2 onto H 2'. The adjoint A* of A - a bounded operator from H 2 into H I defined -
by
(fl,A*f2) = (Afl,f 2)
for arbitrary
fl e H 1 and f2 e H 2
then satisfies A* = A*E 2.
Since
both
H I and
!
H 2 are
separable,
U : H I + H 2 with range UH 1 = Hi; HI) , and UU* = E 2. H1,
so
that
Both U*A
T e B(H1) 1
isomorphically
into
U(U*ATA*U)U* = E2ATA*E 2 = ATA* zero,
moreover,
on
the
is an isometric
operator
i.e., U*U = 11 (the unit operator on
and A'U,
implies Hi,
there
then,
are bounded
U*ATA*U e B(H1) I.
the
last
operators
Since
relation
U
maps
implies
on HI
that
' Being is a trace class operator on H 2.
orthogonal
complement
of
H2, '
this
operator
indeed belongs to B(H2) I.
With
X e B(H2)
arbitrary,
this
operator,
too,
is
then, X.ATA*
zero
also belongs
on the orthogonal
trace can be evaluated with an orthogonal basis H i of H 2. We then obtain
to B(H2) I. Since
complement
of Hi,
its
{f~} of the subspace
51
tr 2(X.ATA*) = i (f~, XATA*f~) = ~ (UU*f2, XATA*UU*f 2) i i = I (U*f~, U*XATA*U(U*f~)) = trI(U*XATA*U) i because
UU*f
i = E2f 2i = f2'
and
{U*F~}
,
is an orthogonal basis
in H I
Both U*XA and A*U belong to B(HI). Therefore U*XAT belongs to B(HI)I, and may be interchanged with A*U under the trace, which finally leads to trI(A*UU*XAT ) = trI(A*E2XAT ) = trI(A*XA.T ) , q.e.d.-
After
these
preparations,
we
can now also
prove
the
first
part
of
Theorem I. For this it suffices to show that ~* is of the form (3.3), with suitable operators ~ diately
implies
Eq.
(k e K) satisfying (3.1), since this imme-
(3.2)
for
~.
Indeed,
once
the
representation
(3.3) for ~* is proved, we know from the previous discussion that ~*, besides being the adjoint of the operation ~, is also the adjoint of the
mapping
adjoint,
defined
however,
by Eq.
(3.2).
two mappings
By
of
the definition
B(H) 1 into
itself
(2.25) with
of
the
the same
adjoint must be identical.
The mapping ~* satisfies the assumption of Stinespring's Theorem with B = B(H). Thus, for all X e B(H),
~*X = A* n(X) A
with a bounded H. According of
H
into
(3.9)
operator A : H + H and a representation ~ of B(H) on
to Lemma subspaces
i, n is reduced by a direct sum decomposition H°
and
Hk,
k
e K.
Denote
projection operators on H onto these subspaces H
Ek
the
and Hk, respectiveO
ly, and introduce the operators
by E ° and
52
Ao = Eo A
,
~
= Ek A
(k e K)
(3.10)
mapping H into H, whose adjoints, mapping H into H, are given by
A*o = A* Eo '
with
the adjoint
be
: ~H + H of A.
in H° and Hk ~ H,
contained well
A*
reinterpreted
interpreted
in
this
~
as
way,
= A* E k
Since
the ranges
respectively,
mapping
H
we d e n o t e
(3.11)
into
of Ao and
the
operators
H
or
o
H,
(3.10)
are
may as
respectively.
them by A and Ak, o
If
respectively.
Then we obtain
A*o = A* H ° , i.e.,
ly.
A~ and A T are the restrictions
I n d e e d , A* a s g i v e n by ( 3 . 1 2 )
(3.12)
= A* Hk
of A* to H ° and Hk,
respective-
satisfies
O
(A~fo,f) = (A*fo,f) = (fo,Af) = (Eofo,Af)
= (fo,EoAf) for
all
f
c H O
and
f ~ H,
= (fo,Aof) = (fo,Aof)
and an analogous
relation
follows
for A k.
O
(Remember that (A*g,f) = (g,Af) for arbitrary g c H and f e H.)
Denoting
by
K
an
arbitrary
finite
subset
of
the
index
O
define a c o r r e s p o n d i n g p r o j e c t i o n EK
operator --
o
on H by
~ Ek • keK O
Since E K < Y = ~(I)
,
O
we get with (3.9) A * E v A < A* A = A* ~(I) A = ~*i = F < 1 • 0
On the other hand, we have
set K,
we
53
o
keK
keK
o
k~K
o
o
as follows immediately from the above definitions of i fore
the operators
and A k. There-
A k on H defined above satisfy the condition
(3.1)
of the theorem.
But
then,
Ak @ 0
as
iff
has
been
k e K'
shown
is
at
before,
most
the
index
countably
set
infinite.
K' By
defined
by
definition,
0 implies K "~ = Ek A = 0. Therefore, with E' = E
o
+
~
keK'
Ek
=T-
k~K 'Ek '
we have (7 - E ' ) A =
[ Ek A k~K'
=
[ Ak : 0 ; k{K'
i.e., A = E'A. In (3.9) we may thus replace A by E'A : H and A*
by (E'A)* = A* E'
+
E'H d~.
~'
: H' + H, so that only the subrepresentation
~IH' of ~, acting as o
~k )'
~k (x) E X
(3.13)
keK' on
actually
enters
may be completely ring in Lemma
Eq.
H' = H ° $ ( @ Hk) , keK'
Hk ~ H
(3.9),
orthogonal
ignored.
whereas
the
Or, in other words:
complement
of H'
the index set K occur-
i may be identified with K', and thus assumed to be at
most countably infinite, without loss of generality.
Considering first
the case
of an infinite
index
set K,
simply take K = {1,2 .... }. Define projection operators
we may
thus
54
Then, using the definition of the operators A
and A k and the decompoO
sition (3.13) of ~, we obtain
A* E(n)~(X) E(n)A = A*o ~o (X) Ao +
(3.14)
[ ~ X Ak k
for an arbitrary
X e B(H).
As has already been proved,
on the right hand side converges ultraweakly
the last term
to
AT X Ak k for
n + =,
since
condition
(3.1)
that for n + ~ the left hand
is satisfied.
We want
side of (3.14) converges
to show now
ultraweakly
to
A* ~(x) A = ~*x.
By definition,
E(n ) commutes
with ~(X) for arbitrary X e B(H) and all
n. This implies +n
E(n)~(X) E(n) = E(n)~(X)
since E(n ) ÷n ~ ultraweakly.
arbitrary
the trace.)
ultraweakly,
(The last statement,
tr(E(n )T) for
~(X)
+n
T e B(H)I , immediately
(3.15)
being equivalent to
tr follows
from
the definition
of
Applying Lemma 2 with H 1 = H, H 2 = H, A = A, an arbitrary
T e B(H)I , and E(n)~(X)E(n ) s B(H) in place of X, we obtain trH(A* E(n)~(X) A T) = tr~(E(n)W(X)E(n ) A T A*) By (3.15), the right hand side converges for n + = to tr~(~(X) A T A*) = trH(A* ~(X) A T) , the last
equality
following
As T g B(H) 1 was arbitrary,
again from Lemma 2, now with ~(X) this shows that
A* E(n)~(X) E(n)A as announced above.
for X.
n+
A* ~(X) A
ultraweakly,
(3.16)
55
Taking thus the ultraweak limit of Eq. (3.14), we obtain
~*X = A* ~(X) A = A*o ~o (x) Ao +
~
~
X ~
.
(3.17)
keK If K is finite,
the argument
leading
to (3.14) yields
(3.17) direct-
ly.
The second summand on the right hand side of (3.17) is already of the form
(3.3).
Therefore,
of B(H) 1 into mapping
itself
of B(H)
itself.
this
implies
operator
@o X = 0
E is the operators
Second,
by exploiting
En;
may be represented
X
and
this
term
defines
mapping a normal
of the operation
but
÷
A*o ~o (x) A o •
all
X e B(H):
upper
limit
~o(En) = 0
the spectral
First,
every
projection
of finlte-dimensional
by Lemma
theorem,
i,
and
pro-
thus
#o E = 0.
every Hermitean
X e B(H)
as an upper limit of operators X
combinations
Hermitean
(3.2),
But ~*, as the adjoint
for
(ultraweak)
jection
linear
by
of a corresponding
thus the same must be true for the mapping ~o : X
But
the adjoint
as given
into
~, is also normal;
being
of projection
thus,
by
operators.
linearity,
for
which are finite n This implies ~o X = 0 for
all
X.
Therefore,
finally,
Eq. (3.17) reduces to
I k~K
k ,
which is the required representation
(3.3), and Theorem 1 is proved.-
A
of
slight
vious. and also
but
useful
Consider,
generalization
Theorem
together with ~, an operation
the corresponding an operation,
set of operators ~ ,
non-selective
there exists k e K',
(3.4) hold true. Moreover,
k~K
1 is
immediately
~' complementary
ob-
to ~,
operation ~ = ~ + ~'. Since ~' is or countably
infinite
so that obvious analogs of Eqs.
another
finite
(3.1) to
the condition F + F' = I implies
k~K'
56
A simplified
notation
to be disjoint,
is obtained by chosing
with K u K' = {I,...,N}
to drop the primes on the operators ~
Theorem exist
I':
Given
an index
any
two
or {1,2 .... }. This allows us
with k g K', and we arrive at
complementary
set J = {I,...,N}
the index sets K and K'
operations
~ and
or {1,2 .... }, operators
~', Ak,
there k ~ J,
with
Z ksJ and
two
complementary
Ak : 1 ,
subsets
K
and
(3.18)
K'
of
J,
such
that,
for
all
T g B(H) 1 and X e B(H), ~T =
~ ~ ksK
T ~
,
'T =
~ ~ keK'
T ~
,
~*X =
~ ~ k~K
X ~
,
(3.19)
~'*X -- ~ ~ keK'
X ~
.
(3.20)
and
The corresponding non-selective ~T =
Vice
versa,
operators
given
~,
operation ~ = ~ + #' is thus given by
~ Ak T ~ k¢J arbitrary
k ~ J
,
index
satisfying
$*X =
~ A~ X Ak • keJ
sets
J,
(3.18),
K,
and
K'
as above,
then ~ and
~'
as defined
(3.19) and (3.20) are operations complementary
The
operators
Theorem 1 Theorem
describing
two
given
index
a
complementary
1') are not uniquely
is already another
(or
Ak
in the
set L,
form
given
(3.2) with
of cardinality
not
~ and
~ #',
To illustrate operators less
and by
to each other.-
operation
operations
determined.
(3.21)
Ak,
than that
according according
to to
this, assume k e K. Choose of K,
and an
"L x K" matrix U with matrix elements Ulk , i c L, k g K, which satisfies U*U = 1 (the "K × K" unit matrix) or, more explicitly, U~kUlk' = ~kk' 1eL
for all k,k' ~ K .
57
(The above the
assumption
existence
of
about
the cardinalities
such matrices
U.)
Replace
of K
and L guarantees
the operators
A k by new
operators
h
Ulk % ,
1
L .
k~K
Then
these
operators
also
satisfy
a
condition
of
the
form
(3.1),
because B~ B I = leL =
~ U~k ~ k,k'eK,leL
Ulk,Ak,
I A~Ak=F~I kEK
"
Moreover, a similar calculation yields I B I T B~ = I A k T A~ = ~T leL keK for arbitrary be described
T ~ B(H) 1. Therefore
the given
operation ~ may as well
by the new set of operators BI,
1 e L, which in general
is quite different from the original set of operators Ak, k e K.
(To become completely rigorous, tional infinite the
convergence sets.
sets only,
considerations
However,
representation
the above argument would require addiif
either
L
in order to illustrate
(3.2),
it is sufficient
or
both
K
and
L
the non-uniqueness
to consider
finite
are of
index
or - if K is infinite - to choose the above matrix U such
that it actually modifies finitely many of the operators A k only.)
Some conclusions ly from Theorem
of more physical nature can also be drawn immediatei. First,
this theorem strongly supports the point of
view that not only projection operations E but, more generally, trary operators F e L(H) tor F corresponds
represent effects.
Indeed,
arbi-
every such opera-
to an operation ~ in the sense of Theorem 1. (Take,
e.g., K = {I} and A 1 = FI/2.)
Second, not
Theorem
uniquely
1 proves
specify
the
that, as mentioned operation
~
before,
the effect F does
corresponding
to
it;
or,
in
58
other words, valence note
different
class
that
an
F
effect apparatuses
need
not
operation
~
perform
the
fl and f2 in the same equi-
same
corresponding
operation.
to a
given
To
see
effect
this,
F may be
constructed as follows. Consider an arbitrary decomposition
F:[% ksK of
F
into
finitely
or
infinitely
many
operators
F k > 0.
For
each
k E K, choose an arbitrary operator U k with U~ U k = I (i.e., an isometric operator), and set ~
= U k F~/2. Then, indeed,
~ A~Ak= ksK which
implies
that
the
scribe an operation arbitrariness That,
in
moreover,
operators
~ belonging
this two
seen
different with
the help
A k satisfy
condition
of
sets
the
of
two different
descriptions of
simple
(3.1)
and de-
to the effect F. The large degree of
different
formal
,
construction
this way can really yield two
~ Fk=F kEK
of
operators operations,
the
examples.
operators
same
~
%
is
constructed
in
rather than only
operation,
Consider,
obvious.
for
is
easily
instance,
the
operations ~ and ~ described by K = {i},
A 1 = F I/2
and = {1,2}, with
an
arbitrary
-I 1 / 2 X 1 = (~F) ,
isometric
operator
U,
~,i~.I/2 X 2 = utah) , respectively.
Both
~ and
correspond
to the effect F, but ~ ~ ~ since ~ transforms pure states F 1 / 2f / jIF 1 / 2fl l W = I f >< f I into pure states W^ = I f^ >< f^ I , with ~ thus
being
a
pure
operation
in
the
sense
of
Ref.
transforms the same initial state W = If >< f I into w = ~ I
><
which is not pure unless Uf^ = ei~
I
ufl
,
[5]
- whereas
59
In particular, necessarily and
described
(2.9),
(3.21)
very
cal
fact,
sion
which
with,
The
operations
limited
"ideal
with
the
ant
among
of
them
actual be
formulae
strong
is
the
described
additional
of
(2.9)
by
and
these
(2.9)
(2.8)
(3.19)
to
and K' = {2}. an
procedures
empiri-
for deci-
formulae
-
even
are considered.
are
assumptions.
that
Eqs.
is also
measurements
(2.8)
postulate
and
E are not
formulae
K = {1}
measuring
only "non-destructive"
measurement" help
not
case
A m = E',
(2.8)
effects
reduction"
particular
A I = E,
many
can
to decision
packet
a very
of Eqs.
are
which
if, as done here,
'~ave
J = {1,2},
there
E
the
validity
since
The
by
represent
e.g.,
effects
corresponding
usually
derived
Particularly
import-
a repetition
of
the
E measure-
with
certainty.
^
ment
in
the
new
Measurements the
first
consider
state
W
fulfilling
kind".
In
a selective
yields
this
this
the
result
postulate
sense
operation
(but
are
"yes" called
slightlY
more
# as constituting
first kind of the corresponding
measurements
"of
generally),
we may
a measurement
of the
effect F = ~*i e L(H),
if
^
tr(FW) i.e.,
-- tr(F.~W)/tr(FW)
-- tr(#*F.W)/tr(FW)
if tr(~*F.W)
for
all
But
since
states
W
tr(FW)
= tr(F-~W) W.
= I ,
= 0,
Therefore
with
non-vanishing
= tr(~W) the
= tr(FW) transition
= 0 implies
last
equation
an operation
~W = 0,
must
~ constitutes
probabilities
and
actually
thus hold
also for
an F measurement
tr(FW). tr(~*F.W)
all
states
of the first
kind if (and only if) ~*F = F This
condition,
(For
counterexamples,
unless a
clearly,
practice
unitary
most
not
valid
take K = {I},
F is a decision
suitable
is
.
U,
effect such
measurements
are
E.
for
arbitrary
A 1 = F I/2,
In the latter
that
~*E = EU*EUE
known
not
to be
operations
~.
such
that ~*F = F 2 # F
case,
set A 1 = UE with
~ E.) "of
Since the
first
also
in
kind",
60
there
is thus neither
a practical
nor a theoretical
justification
for
such a postulate.
Not
every
Perhaps bed by
state
transformation
the most
important
of physical
counterexample
interest
is an operation.
is time reversal,
as descri-
[11] @ : W + T W T*
with an antiunitary
(3.22)
operator T. (I.e., T is antilinear, T(af + bg) = a*Tf + b*Tg
,
and satisfies T* T = T T* = i , with T* defined by (f,T*g) = (g,Tf) for all f, g e H.)
By
(3.22),
the mapping
@ has the following
proper-
ties:
(i)
It transforms
(ii) (iii)
pure states into pure states.
It is trace preserving:
tr(@W) = trW.
It
of
transforms
the
set
pure
states
onto
(rather
than
only
into) itself.
If @ were therefore
an operation,
it would be a pure one,
it could be also represented
according
to (i), and
in the form [12]
@ : W ÷ A W A*
with
a
would
linear be
unitary,
operator
isometric: i.e.,
also
needed
to exclude
not
of
be
the
A.
By
A*A = i.
(ii) Then
satisfying
the possibility
simple
form
(3.23)
(3.23)
(the (iii)
"non-selectivity" would
AA* = 1. that [12].)
@,
imply
(Actually although
However,
of
that
A
(iii) being
a state
0),
A
is even is
pure,
also might
mapping
of
61
the
form
(3.23)
(3.22)
with
operation. ly,
but
with
antiunitary
unitary
Or,
A
Thus,
in other words:
in principle
an arbitrary
[ii].
-
ensemble
T can not
be
indeed,
rewritten
time
it is impossible
to build
an apparatus
of microsystems,
reversal
is
form
not
an
- not only practicalwhich,
would
in the
turn
when
applied
to
it into an ensemble
in the time reversed state.
The mapping
(3.22)
of K(H)
into
ly to B(H)I,
and is then given by
itself
may easi!y b e e x t e n d e d
@ : T + T T* T*
This
extended
tion, ve,
otherwise
also
2-positive), purely here.
is
But
it would
this
directly
and
such
proof
mathematical
point
of
and
positive,
by
construc-
then it can not be completely
correspond
prove
(3.24)
.
complex-linear
and trace preserving.
since
could
mapping
linear-
(e.g.,
might view,
even we
to an operation. by proving be feel
of
that
some
no
need
Although
@ is not
interest to
positi-
from
present
we
even the this
62
§4 Composite Systems
With the first representation theorem we have obtained a purely mathematical
- although
comparatively
simple,
- description of arbitrary operations
and thus practically
useful
~. In §5 below we shall present
a completely different description in the form of our second representation this
theorem. theorem
Although
can
be
interpreted
model
of
the measurement
rent
from
the physical
tation of
requires
composite
somewhat
in
process, point
This
complicated
terms
and
is
of view.
some knowledge
systems.
more
of
thus
Its
a
mathematically,
quantum
perhaps more
formulation
touched
already
transpa-
and interpre-
of the quantum mechanical
subject,
mechanical
description
in
§2
and
§3,
shall thus be discussed now in a more systematic way.
Given
two
distinct
and
nonlnteracting
quantum
mechanical
systems
I
and II with state spaces H I and HII , respectively,
the state space of
the combined system I + II is the product Hilbert
space H = H I ~ HII.
Density
equivalence
operators
W c K(H),
therefore,
describe
of preparing instruments ~ for pairs of mlcrosystems,
classes
each pair consi-
sting of one system I and one system II, and effects F c L(H) are to be
measured
by
effect apparatuses
~
exposed
to,
and
possibly
trig-
gered by, such pairs.
Consider
an
arbitrary
decision
effect
system I. Standard quantum mechanics sion effect E 1 ~ Iii ~ L(H) ing
physical
aparatus
interpretation.
e I which,
the effect composite
El,
when
of
tested,
e.g.,
system
II.
Assume to
can also
(This
by preparing
the
isolated
associates with it another deci-
there exists the
isolated
be applied
property
two ensembles
of
at least one effect system
I, measures
to subsystem I of the
system I + II, without being influenced
presence
of
of the composite system, with the follow-
applied
but which
E 1 c L(H I)
the
in that case by the apparatus
could
be
of pairs I + II in the same
63
state
W,
removing
comparing these eI
subsystem
the probabilities
two ensembles.)
has
II
to
system;
be
for
as
an
corresponding
each
the
When applied
considered
the
of
pair
in
triggering
to pairs
effect
effect
one
of
in
the
I + II,
apparatus L(H)
ensemble,
and
apparatus
in
the E I apparatus
for
is
the
composite
assumed
to
be
E1 = E 1 ~ IIi. Likewise,
an apparatus
II while
being
ell measuring
insensitive
a decision
to system
effect
I yields,
when
Ell on system applied
composite system I + II, the decision effect EII = 11 @ E I I
For arbitrary commute.
Therefore
mechanics, these
sense.
(See
different
gers
be used in
and
both pointers
"El and E11
exists
suitable
pointers",
both
a
and
the
times,
(Ell)
respectively.
the yes-no measurement
in
to
of N pairs
second
composite the
apparatus
responding
that an ensemble
E 1 and Eli
single
the
quantum system;
following with
two
systems
in a state W trig-
pointer
This
approximately
apparatus
can also
"El and Ell" , defined - as
algebra)
-
are triggered
together
by the same single pair I + II.
associates
with
to give
this
the
result
correlation
"yes"
if
measurement
the product
(4.1)
El-E11 = (E 1 @ Iii)(I I @ Ell) = E 1 @ Ell of
on H
(Boolean
mechanics 11
the
There
tr(EIIW)N
logic
of
together",
(El)
to perform
effects
the
in L(H).
to standard
"measured
§6.)
be
operators
again according
decision
can
such
first
ordinary
Quantum
also
"yes-no
the
tr(EIW)N
effects
I + II,
projection
they describe,
"commensurable"
i.e.,
(pairs)
E 1 and Eli , the
to
operators
El
and
Eli,
i.e.,
another
decision
effect
EI,I I = E 1 ~ Ell of the composite system.
In the particular case considered, ment
an apparatus
for the joint measure-
of E 1 and Ell can be realized as follows. Assume
there exists an
64
apparatus
e I measuring,
as above,
the effects
E 1 on isolated
systems
I and ~I on composite systems I + II, and which is not only insensitive
to,
but
latter
case.
subsystems not
also
does
(In
view
I and II,
implausible.
discussed
below.)
apparatus
influence
of
the
the existence operation
One
then expects
an
~I and ~II
of
the
subsystem
interactions
by
such
Ell
apparatus
ell
the
is at least
an
apparatus
that a combination
on the composite
II in the between
of such an apparatus
performed
analogous
together
at all
absence
The
e I with
measures
not
is
of such an E 1
for
system.
subsystem
This
II
is indeed
confirmed by the detailed theory presented below.
A particular as
follows.
one~
type of state W of the composite Take a preparing
Wll , for
combine
system
II.
instrument
Prepare
them into N pairs
view
explicit ~I
and
of
this
preparing
construction ~II'
it
"~I and ~II"
(Eq.
is
N systems
with
each apparatus,
and
procedure
intuitively occurs
of
state.
of an apparatus
(4.1))
w I for system I and another
I + II. The state W of such an ensemble
N pairs is called an uncorrelated
In
system can be prepared
for
W
and
the
above-described
for the combined measurement
obvious
that
the
in an uncorrelated
decision
of
effect
state W with
the
probability
(4.2)
tr((E I ~ Ell)V) = tr(EiWl).tr(EllWll ) ;
i.e.,
the
probability
for
the
joint
occurrence
of
the
effects ~I and ~II is the product of the probabilities
"subsystem"
of the effects
E 1 and Ell in the subsystem states W I and WII prepared by the instruments
w I and Wll,
coincide composite
with
the
respectively.
The
probabilities
of
two last-mentioned the
effects
~I
probabilities
and
~II
in
the
system state W since, with either Ell or E 1 replaced by the
unit operator,
(4.2) implies
tr(EIW) = tr(EiWl)
,
tr(EllW) = tr(EllWll)
.
(4.3)
65
Like
(4.2),
these relations
also have a very
simple and intuitively
obvious physical interpretation.
Eq.
(4.2)
implies
that
the
density
operator
W
of
an uncorrelated
state is the tensor product
= of the density of
operators
which
W
of
(4.2)
side
is
W I @ WII
W I and WII describing
"composed". is
This
identical
trI(E I @ EII)(WI @ WII) ) . EI = i fI >< fI; and E I I =
(4.4)
to,
For
is and
the subsystem
because can
thus
one-dimenslonal
the be
states
right
hand
replaced
by,
projection
operators
i fii >< fiI ] , for which
E1 @ Ell = I fl @ fll >< fl @ fi1 I ' (4.2) can then be rewritten, with
=
w - w I @ wii
in the form (4.5)
((fl @ fll )' ~(fl @ fll )) = 0 • This
is true
restriction with
for arbitrary to unit
suitable
(1.13)
scale
for A = 8,
unit
vectors
vectors
is easily
factors.
Applying
f = fI @ fII'
fl e H I and f l l e
removed now
by multiplying
the polarization
The (4.5)
identity
and g = gI @ fII' with gI ~ HI arbi-
trary, we obtain from (4.5)
((fl ~ fll )' ~(gl @ fll )) = 0 • A similar identity,
HII.
66
4((fl ~ fll )' ~(gl @ gll ))
=
((fl ~ (fll+ gll ))' ~(gl @ (fll+ gll )))
-
((fl ~ (fll- gll ))' ~(gl 0 (fll- gll )))
+ i((fl ~ (fll-igll))' ~(gl ~ (fll-igll)))
- i((fl 8 (fll+igll))' ~(gl @ (fll+igll))) ' then leads to ((fl @ fll )' ~(gl @ gll )) = 0 , valid
for
all
combinations
fl' gl g HI
and
fll' gll e HII.
Since
finite
linear
of product vectors are dense in H, this finally implies
A = 0, i.e., Eq. (4.4).
Consider
now,
more
generally,
an effect F I s L(HI)
system I which need not be a decision effect. the particular
case F I = EI,
of the isolated
Assume,
that there exists
as above for
at least one effect
apparatus fI in the equivalence class F I which can also be applied to composite
systems
system
in
II
measures
this
I + II, case.
without
affecting
If applied
in
nor
being
this way,
affected
by
the apparatus
fI
a certain effect ~I e L(H) of the composite system. By this
operational
definition,
the effect ~I must occur in an uncorrelated
state (4.4) with a probability
(4.6)
tr(Fl(W I @ WII)) = tr(FiWl) ,
coinciding with the probability for the effect F I in the state W I of subsystem to
I. Likewise,
effects
FII
of
suitable the
effect apparatuses
isolated
subsystem
II
fII corresponding define
effects
~II e L(H), with
tr(Fll(W I @ Wll)) = tr(FllWll) ,
(4.7)
67
when applied to the composite system. Finally, such apparatuses together.
fl and fll is again expected to measure ~I and ~II
(Thus ~I and ~II are "coexistent"
detail in §6.) The same combined apparatus measure
the combination of two
the effect ~I,II = "~I and ~II"'
if and only if both
effects,
as discussed in
can then also be used to
defined
fl and fll are triggered
single pair I + II. Again the probability
(as above)
to occur
together by the same
for ~I,II'
i.e.,
for the
joint occurrence of ~I and ~II' must factorize in the form
(4.8)
tr(Fl,ll(W I @ WII)) = tr(FlWl).tr(FllWll) for
uncorrelated
states.
Inserting,
in
particular,
pure
states
WI = ~fl >< fl I and WII = I fll >< fi1 I , Eq. (4.8) now implies (4.9)
~I,II = FI ~ FII ' while Eqs. (4.6) and (4.7) imply
~I = FI 8 III '
(4.i0)
~II = ii 8 FII "
(Compare the derivation of (4.4) from (4.2), and note that (4.6) and (4.7) are just particular cases of (4.8).
See also Eq. (3.8) of §3.)
We have thus shown that relations which are already known for decision
effects
form
of
Effects
(e.g.,
Eqs. of
the
(4.1))
(4.9)
and
form
can be
(4.10),
(4.10)
and
generalized
to
arbitrary
(4.9)
will
immediately, effects
be called
FI
in the and
FII.
subsystem
and
correlation effects, respectively, in the following.
For an arbitrary state W e K(H) of the composite system there exists, as is well known, a unique density operator Trll ~ e K(H I) for subsystem I, called the reduction of the state W to that subsystem,
such
that tr((F 1 O
iIl)~)
= tr(Fi.Trll ~)
(4.1i)
68
for all F I e L(HI).
Thus Trll ~ describes the statistics of arbitrary
subsystem I effects in the given state W; or, in other words, describes
the state of the N subsystems
Trii ~
I in an ensemble of N pairs
I + II in the state W.
More
generally,
there
exists
for
any
T e B(H) 1
a
unique
operator
T I = Trii ~ g B(HI)I, which is defined implicitly by requiring
(4.12)
tr(Xi. TrllT) = tr((X I @ Iii)~)
for
all
X I e B(HI).
(4.12)
defines
a
Namely,
linear
for
fixed
functional
T,
the
T(X I)
on
right
hand
side
B(HI) , which
is
of
norm
continuous since
IT(XI) I < ;IXI @ llili IIT;I1
=
IITII1 llXll; ,
and therefore is of the form tr(XiTl) with a unique T I c B(HI) I. Eq. (4.12) thus defines a mapping Trll
: B(H) 1 + B(HI) 1 which, obviously,
is
since,
linear.
It
tr(Xi. Trll ~) > 0 plies
is
also
for
positive
all X I > 0,
TrlI(T*) = (TrlIT)*;
i.e.,
and
by
(4.12),
thus Trll ~ > 0.
T > 0 This
Trll is a real mapping,
implies also
and
im-
(4.12)
with X I = 11 yields
tr(Trll ~) = trT;
i.e., Trll is trace preserving.
Therefore
states
into
(4.11)
Trll
maps
W g K(H)
states
Trll ~ e K(HI) , and
follows as a special case of (4.12). For T = T I @ TII , (4.12)
implies tr(Xi-TrllT) = tr(XiTl).trTll , and thus TrlI(T I @ TII) = trTll. T I
(4.13)
for arbitrary T I e B(HI)I, TII s B(HII) I. This also implies that Trii maps B(H) 1 onto B(HI)I, and K(H) onto K(HI).
A more
explicit
X I = I fi>
equation for Trll ~ follows
with
an arbitrary
unit
vector
from
(4.12) by inserting
fI s H I . Evaluating
the
69
trace on the left hand side with a particular orthonormal basis {f$} i in HI, for which fl = fl' and the trace on the right hand side with i k i the basis { f I @ g l I } i n HI @ H I I , w i t h { f i } as b e f o r e and a n a r b i t r a k ry basis {gll } in HII , we then get from (4.12) k (fl' Trill fl ) = ~ ((fl @ gll )' k By
multiplying
with
scale
factors
and
T(fl ~
k gll )) "
applying
the
polarization
identity (1.13), we obtain from this the well-known formula k (fl' Trill gl ) = ~ ((fl @ gll )' k
T(gl ~
k gll )) '
(4.14)
valid for arbitrary fl' gl e H I . Thus Trll ~ results from T by performing a "partial trace" with respect to HII; hence our notation.
By interchanging
the rSles of H I and HII , we obtain another mapping
Tr I : B(H) 1 ÷ B(HII)I,
the
partial
trace
with
respect
to
HI,
with
analogous properties and a similar physical interpretation.
An effect apparatus uring
the
effect
fl' performing a selective operation ~I by meas-
F I = #~i I on subsystem
I and not
interacting with
subsystem II, can also be used to perform a selective operation ~I on the composite system I + II. In this case, pairs I + II are selected according
to the occurrence of the subsystem I effect ~I = FI @ III'
as measured by fl when applied to such pairs. If #I is represented in the form (cf. Theorem i)
~IWI = ~ Alk W I A~k keK
with suitable operators Alk (k e K) on HI, the operation il is given by
~i~ = ~ (Ale @ Iii) W (Ale ~ Iii)* . keK Then, indeed, the effect corresponding to ~I is
(4.15)
70
Z (Alk @I i i ) * (Alk 0 i l l ) = ( Z A~k Aik) 0 1II = FI @ III = FI '
keK
keK
in accordance with (4.10), whereas for uncorrelated states we have
(4.16)
~l(Wl 0 Wll) = ( [ AIkWIA~k ) @ WII = ¢iWl O Wll , keK as expected form the operational definition of ~I"
(Compare also Eqs.
(3.6) to (3.8) of §3.) One can also show that,
conversely,
every operation ~I which acts on
subsystem I of the composite system I + II only, and therefore transforms
uncorrelated
states
be of the form (4.15).
in accordance
with Eq.
(4.16),
The proof is rather lengthy,
indeed must
however,
and is
therefore omitted here.
With ~I as given by (4.15) and arbitrary X I e B(HI) and XII e B(HII) , we have ~ ( X I O Xll) = ( Z A~kXIAIk) @ Xll = ¢~X I 0 Xll • keK From this and Eq. (4.12) we get, for arbitrary W e K(H), tr(Xi-Trii(!IW)) = tr((X I ~ Iii)'~i ~)
= tr(!{(X I 0 I i I ) . ~) = tr((@~XI 0 l i i ) . ~) = tr(~Xi.Trll ~) = tr(Xi.@i(TrliW))
•
Since X I is arbitrary, we finally obtain
TrIi(~i ~) = ¢I(TrIIW) This equation means, after
physically,
(4.17)
•
that the final state of subsystem I
the operation iI - as given,
up to a normalization
Trii(~i ~) - may also be obtained by applying
factor,
by
the operation ~I to the
71
initial state Trll ~ of subsystem I. By the operational definition of ~I'
this must
composite
indeed be true for arbitrary
system.
(For uncorrelated
initial states W of the
states,
this
property
of ~I
is
already expressed by (4.16).)
The measurement
of the correlation effect ~I,II = "~I and ~II" in an
ensemble
of N >> 1 pairs
combined
aparatus,
acting
on
I +
II in a given state W by means
consisting
subsystem
of
fI
and
a
similar
II, may now also be described
of a
apparatus
flI
as follows.
The
apparatus fI is triggered by N I = N tr(Fl ~) = N tr(~l ~) pairs, which form an ensemble in the new state = IiW/tr(~IW) In this ensemble,
•
the subsystem II effect E11 = 11 0 FII measured by
fll then occurs NI,II = N I tr(!ll~) = N tr(Fll-~l ~) = N tr(i~Fll. ~) times.
Thus
fl
and
fll
are
triggered
together,
in an arbitrarily
given state W, with the probability NI,II/N = tr(Fl,ll.W) , where
~I,II = i ~ l l
=
[ (Alk 0 111)* (i I 0 FII) (Alk 8 iii)
keK
= ( ~ A~k Alk) O FII = F I O FII , keK in accordance with (4.9).
This
interpretation
parts
of
the ~I,II
measurement
is appropriate
if the
fI and fII of the combined apparatus are applied to, and pos-
sibly triggered by, each single pair I + II in the order considered (i.e., applied fII
first before
fI'
then
fli ).
If,
fI' a completely
interchanged
again
leads
to
vice
versa,
analogous (4.9).
the apparatus
fII
is
consideration with fI and Our
previous
derivation
of
72
Eq.
(4.9)
from
the
requirement
and fII in uncorrelated cular
(4.8)
for the joint
states is more general,
also independent
of the temporal
order
triggering
of fl
however, and in partiof the two measurements
performed by fI and fII"
According
to
of subsystem is not
(4.16),
the
operation
II when applied
true for more
~I does not change
to an uncorrelated
general
states W;
the state WII
state W I @ WII.
in fact,
one may
This
easily con-
^
struct
examples
for which
the final
state TrlW of subsystem II, with
^
= ~iW/tr(~iW), no means
is different
surprising.
from its initial
state TrlW.
If in the state W there are correlations
the subsystems
I and II, then a selection according
effect
indeed
~I
changes
may
are
thus
This is by
not
change due
the
state
of
to an interaction
to a subsystem I
subsystem between
between
II.
Such
state
the apparatus
fI
and subsystem II, but rather result from the interplay of the correlations
and
be absent
the applied
if no selection
sponding non-selective
According
selection
to Theorem
the non-selective
procedure.
is made,
i.e.,
Therefore
they must
also
if instead of iI the corre-
operation ~I is considered.
1' and an obvious
operation
~I
generalization
performed
by
of Eq.
the apparatus
(4.15),
fI on the
composite system transforms W into the new state
~I ~ = ~ (Alk @ Iii) ~ (Alk @ Iii)* • kcJ
Here J is an index set of which
the set K in (4.15) is a subset, and
the operators Alk (k g J) on H I satisfy
A~k Alk = 11 ksJ For arbitrary subsystem II effects ~II = II @ FII we then get ~(i
@ FII) = ~ (Aik @ iii)* (i I ~ FII) (Aik @iil) ksJ
= (k~J ~ A~k Alk) @ and thus
FII
= 11 @
FII '
73
tr((l I @ FII)'~I ~) = t r ( ~ ( l I @ FII)" ~) = tr((l I @ FII)" ~) •
This means (compare Eq. (4.11))
Tri(~i ~) = TrIW ; i.e.,
the reduction
of the composite
system state to subsystem II is
indeed unchanged by the operation ~I"
The
preceding
ternal
consistency
without the
discussion of
interaction.
theory
to
should be sufficient the
For
composite
formal
later
description
use,
systems
to demonstrate of
however,
I +
II
we
whose
the in-
composite have
to
subsystems
systems
generalize I and
II
interact with each other.
The above
operational definition of subsystem and correlation effects
can not be generalized stems,
since then,
immediately
obviously,
to the case of interacting
one can not apply effect apparatuses
one subsystem without perturbing the other one. Likewise, lity
of preparing
the subsystems
subsy-
uncorrelated
states
becomes questionable.
by independent
to
the possibi-
preparations
In order to circumvent
of
difficul-
ties of this kind, we restrict our attention to so called "scattering systems",
which are similar enough to noninteracting
ted explicitly
in a simple way,
ones to be trea-
and which on the other hand are just
the kind of systems encountered later on in §5.
A
quantum
system,
mechanical
if it behaves
site
system
well
be
prepared
were
no
interaction
there would (4.10)
and
of
system
is
exist (4.9))
Given an arbitrary
here
a
binary
scattering
in both the distant past and future as a compo-
two nonin=eracting and
called
studied at
all,
particular with
the
subsystems
independently the
state
subsystem physical
of
each
space would
other.
interpretation system,
If
can as there
be H I @ HII,
and correlation
state W of the composite
to measure the probabilities
I and II, which
effects
discussed
and
(Eqs. above.
it would suffice
tr(Fi,iiW)__ for sufficiently many correla-
74
tion
effects
(4.9)
F I = I f I >< fi I
in
order
and
to
determine
W
completely.
(Take,
so
tr(Fi,iiW)
FII = I fii >< fii J,
that
((fl @ fII )' ~(fI @ fII ))' and recall the derivation
The
state
fied
space
with
effects
H I @ HII ,
of
the
"subsystem" Given
form
an ensemble
system
and
-
I
also
-
with
of such systems
of a binary
and
II.
following
most
cases
other
for
them.) the
the
t < T_,
Such
form
measured ciently
(i.e.,
separate
(4.10),
depend,
many
measurements
seem
seem
mes,
in the given ensemble.
now
this
described
effect type
separated
separate
occurrence
of
the
(In
form each
measurements
on
by
of
operators
of two such effects
again
FI
conventional however,
here,
ensemble
of
from
form
uniquely
however.
scattering
the very
at all could be performed
Second,
many
to arise
systems
even to and
tween the preparation
In
the
for one subsy-
(4.9).
when
Suffi-
determine
First,
the
the state W
systems)
might
at a time later than T_, so that it describes
impossible
ciently
are
to an
of
corresponding
of interacting
action.
on
operator W on H = H I @ HII.
been prepared ble
systems
facilitates
joint
corresponds
difficulties the
general,
constructed
II are spatially
clearly
measurements
in
the other one can thus also be applied,
I and
the
before which
will
T_
measurements
while
together
statistical
Certain
which
- by
time
scattering
subsystems
exists
like a pair of two noninteracting
with
to all
the
meaning.
behaves
stem and not
t < T_,
there
system - a time T
Effect apparatuses
at times
"correlation"
physical
in some state W,
state W of the ensemble. interacting
be identi-
for FII = Iii or F I = Ii,
the
scattering
This
particular
=
(4.4).)
system may still
exist
including,
(4.10)
of the ensemble
subsystems
scattering
there
(4.9)
effects
the definition each
H of a binary
of Eq.
e.g.,
if W was
measure FII
prepared
correlation
in
the
beginning,
before
and therefore
effects
the
mechanics
that a given
effect
one
more
F I @ FII
limited
or
no
time T , it might
time
and the onset of the interaction
quantum
an ensem-
the onset of the inter-
before
remaining
have
less
F can be measured
with
suffi-
interval
be-
at time T .
explicitly
assu-
"in principle"
by
75
many
different
apparatuses
f including,
in particular,
one apparatus
"operating" in an arbitrarily small time interval around an arbitrarily given
time
t.
characteristic
Consider,
function
e.g.,
Xv(Xo)
a free
of
its
particle,
position
t = O. Here V is a given spatial volume,
and
take for F a
operator
X ° at
time
and its characteristic func-
tion XV is defined by
×v(.) =
Then
XV(Xo)
is
a
projection
i
for
x e V ,
0
for
x ~ V .
operator,
thus
describing
a
decision
effect E. The simplest apparatus measuring E (at least approximately) is a counter time
occupying
interval
around
the volume V and "switched on" during a small t = O.
The
solutions
of Heisenberg's
equations
of motion for position and momentum of a free particle, 1 Xt = X o + m rot ' imply, however,
Pt = Po '
that E may also be represented as i E = XV(X t - ~ Pt t)
in
terms
apparatus
of
measuring
"operating" used
to
position
X t and
this
momentum
particular
at (or at least around)
measure
E.
Although
should also be measureable corresponding
apparatus
this
at time
can
not
be
Pt
at
function
any
other
time
of X t and Pt'
t.
An
and thus
the time t, might thus as well be indicates
that
"in
principle"
t, the actual construction deduced
theoretically,
E
of the
and might
in fact be quite difficult in practice.
An
assumption
removes
the
of
this
second
type,
if made
difficulty
for
mentioned
the
subsystems
above,
since
gous
assumption
for
preparing
instruments:
W, there shall exist instruments w preparing at
arbitrarily
times.
prescribed
(in
particular,
II,
then arbitrary
subsystem and correlation effects can indeed be measured time interval before T . The first difficulty
I and
in any given
is removed by an analo-
Given
an arbitrary
state
this state and operating at
arbitrarily
early)
76
In order do
not
to prevent assume
any sense. view
when
the
theory.
duration
one
a
included ment;
plies
the
other
at
mention
theory,
words,
use
or
of
ned
in space,
count
all
present
ted
It is also unnecesalways
time
of
the
preparing
are
here
treated
e.g.,
type
at
other
a
the
as
here;
is
be
to
instru-
construction
different.
described
above
also.
but
(This
see below.)
places
time or
or measuring same
a
counter,
application
the
the
have
im-
We should enter
Without
the such
to understand
how arbitrary
be measured
in arbitrary
states
even
F
and
states
more
care,
and measuring experiment
To
apparatus
subtleties
in terms if
one
a
e.g.,
goes,
however,
into
occupy
classes
account
actually
earlier
which
the very
equivalence
cannot
they
operates
theory
of
takes
instruments
since,
present
W
W. But
the
that
be combi-
same
region
than the preparing
properly
takes
into
ac-
far beyond the scope of the
discussion.
states in
in terms of the correlation
W of
the
above
of
Any
such
for
for WII.
operating
at
Eq.
state
w I for This
-
the
may
state
procedure
before
and also
W I of
preparing onset
some particu-
turn out to be uncorrelato
be prepared
requires
the
(4.9),
therefore
subsystems,
of particular
times
effects
system will
(4.8),
noninteracting
Wll,
existence
the scattering
sense
instrument
state W.
The
assumptions
requires
ring
the
in
"sensibiliges"
of
explicitly,
be quite unrealistic
processes
instruments
of
in
be difficult
etc.
= W I @ WII. bed
of
"instantaneous"
application;
picture
or the effect
When analyzed lar
times
effects
these
plate.
be
already
single
instrument,
of
accelerator,
two
less
preparing
in a
such
the Heisenberg
it would
instruments
certain
an
to
processes.
time
specification
F could
defintion
on"
that
more
assumptions effects
definite
to stress here that we
in fact,
Nevertheless,
a
different
that we
would,
and measuring
photographic
in
in
applied
and
"switches
"exposes"
also
preparing
we want
or measurements
Such an assumption
for
finite
of
preparations
of actual
sary
misunderstandings,
i.e.,
be
by
in the way descri-
by means
subsystem
described
I and
of a prepaanother
one,
the use - and thus presupposes instruments of
w I and Wll , both
interactions
in
the
given
77
Up to now we have considered effects
(4.10)
and
(4.9)
the "incoming"
only.
in a given state W behaves subsystems say, and
in
the
on can also correlation
(Like
T_,
guous
to some
distant
But
T+ also
future
now
depends
extent,
a binary
scattering
system
like a pair I + II of two noninteracting
operationally
effects,
since
subsystem and correlation
as well,
later
define analogous
to be measured
on the state W.
and may
not
even
than
some
"outgoing"
at times
T+
subsystem
later
Both T+ and T
exist
time
than T+. are ambi-
in a literal
sense
if
the interaction does not vanish exactly but only becomes "nondetectably weak" states
for
"sufficiently"
early
T_ and T+ will satisfy T
well-defined
time
interval
set T_ = ~
Being ing
thus defined
times
(4.10) will before
Although
interaction,
there
interaction at all,
may
in a different
way,
or joint measurements do
not
triggered and
the
after
coincide
in the
physical
analogous, ing
a
with
given
the
state
interaction;
situations
corresponding
correlation
mathematically,
W
with
these
before
incoming
different
two
that T
the "in" analog T became
Therefore,
of T+,
probabilities
the
therefore
since nevertheinteraction
are
in such a way,
that
the
(4.9) and (4.10). This "identification" -
an isomorphism T+ of H onto H I ~ HII - can not be the
the
which led to the represenation Actually,
of T+,
(4.9)
we have not introduced
but simply assumed H E H I ~ HII
identity map),
in order
T+ becomes an isomorphism
same space H I ~ HII; nition
in general
measurements
after
effects
the state space H of the scatter-
and (4.10) for the incoming effects. explicitly
and
describ-
and subsystem effects are described by operators
same as the one used previously,
tation.
exist
at sufficiently
fl' for instance,
the tensor product H I ~ HII
of product form, as in Eqs.
(so
also
the outgoing effects
on the subsystems
one can also "identify"
system with
outgoing
or less
for which one could
can not be described by the same operator F. However, less
for most
thus including a more
or (4.9). A subsystem I apparatus be
times.
and T+ = -~.)
separate
late
late
< T+,
of
states without any detectable
and
to simplify
our no-
of H = H I 0 HII onto the
i.e., a unitary operator S on H I ~ HII. By defi-
the outgoing
correlation
effect
defined
in analogy
to
78
(4.9)
is
represented
on
T+HH = H I @ HII
by
the
product
operator
F I @ FII. If transformed back to H, this operator acts like T~I(F I @ FII)T + = S*(F I @ FII) ~ • Outgoing
correlatKon
effects
are
therefore
represented
in our
state
space H by operators of the form
I,II Four which
= ~*(FI @ FII)~ '
(4.18) in
from the corresponding incoming e f f e c t s
differ
given by (4.9) by a fixed unitary transformation.
~I,II ~ ~I,II
as
Likewise,
the outgo-
_out = S,(I I @ FII) ~ . ~II
(4.19)
ing subsystem effects corresponding to (4.10) a r e
FOUt = S,(F I @ iii) ~ I ' The
operator
is nothing
S,
else
representing than
the
the usual
overall
effect
scattering
of
the
operator
interaction,
or S matrix.
Its
knowledge permits the prediction of the probabilities - out tr(!l,ll ~) = tr(S*(F I @ FII)~ ~) for
arbitrary
outgoing
correlation
(and
(4.20)
subsystem)
effects,
the state W is also known - e.g., from the measurement ly
many
been
incoming
prepared
(Predictions
correlation
as
a
known
effects,
or
uncorrelated
of this kind are usually
from
state
the in
of sufficient-
fact the
provided
that
W
distant
has
past.
cast in the form of scattering
cross sections.)
The given
quantum here
particular
mechanical is
rather
types
ill-defined
-
of
both
the system during Nevertheless,
this
description
incomplete, yes-no
of
and
the time interval of
a
binary
course.
measurements
physically
type
of
We
only,
scattering have
which
mathematically
-
discussed moreover when
is
sometimes
quite
and sufficient - e.g., in scattering theory, or here in §5.
very become
applied
of interaction between T
description
system
to
and T+. adequate
79
Quantum mechanics
in the form presented here uses the so called Hel-
senberg
Indeed,
given
picture. state
Moreover, the
is
described
the complete
at different
corresponding time
is
here
characteristic by
a
specification
time of application;
instrument
ry
as
fixed
of
that
picture,
statistical
operator
of an effect apparatus
yes-no measurements
a W.
includes
performed with the same
times are thus considered
as different.
(The
operators F are usually related to each other by unita-
translation
operators.)
The
operator
S
introduced
above
is
thus the Heisenberg picture S matrix.
If one discusses only measurements outgoing subsystems
on the noninteracting
of scattering systems,
the Dirac or interaction pictures picture,
one
rewrites
incoming and
as we are doing here, then
is also quite appropriate.
the probabilties
(4.20)
for outgoing
In this correla-
tion effects in the form W out) tr((F I @ FII) S W S*) = tr(Fl,l I _ with W °ut
Accordingly, fects
by
=
S W
S*
.
(4.21)
one describes both incoming and outgoing correlation ef-
the
same
operators
~I,II
given
by
state (i.e., a given preparation procedure)
(4.9),
whereas
a given
is described by two diffe-
rent statistical operators: by W in E W if used
to calculate
tr(Fl,llWin),
probabilities
and by wOUt,
abilities
tr(Fl,ll ~°ut)
considers
an
same", state and
and
explains
change II
incomig
in
measurements,
_W in
the
their
W °ut + _ time
for incoming
effects
in the form
as given by (4.21), which yields the prob-
for
and
(4.22)
outgoing
the
effects.
corresponding
(in general)
due
outgoing
different
to the interaction
interval
In a sense,
separating
effect
one
thus
as
"the
probabilities
between
"incoming"
by a
subsystems
and
I
"outgoing"
80
This description has the formal advantage tlon effects the
interaction.
ver, of
have
the same simple product form both before and after (They
in the interaction
spoiling
the
that subsystem and correla-
unique
still
do not make
sense
operationally,
interval.)
But
assignment
of a statistical
howe-
this is achieved at the price operator
W
to a
n
given preparation procedure - i.e., at the cost of conceptual simplicity.
From
appealing,
this
point
o f view,
the
Schr~dinger
picture
and shall thus not be described here at all.
is even less
81
§5 The Second Representation
We are now ready cal
to derive,
representation
most
easily
of
obtained
as announced
arbitrary from
a
and the corresponding
Assume
for
this
measurement
can
space by
purpose also
is denoted
the
be
by Ha,
effect
described
quantum
statistical
The microsystem,
is
described
in
a
state by
space
the
the
other,
and are independently
ly.
microsystem
Here
Wa
is
measurement", tion
of
the
be
chosen
act
with
will
each
other
on
artificially),
microsystem. by
the
W and
The
on
(e.g.,
with
the microsystem. operator
such
a
scattering
from
f
or
by
does
This
before.
the microsytem
W and Wa,
and
f
with
is A in
each
respective-
is
"ready
for
not
a
the microsystem the apparatus
interval is
each
(which
"switched
other
of
on
or
may
inter-
in general and
afterwards
"absorb"
off"
if - as
"destroy"
the
the
effect
measured
by a certain
yes-no
measure-
pointer")
measurement which
marked
after
is described
is again
fixed
by
the
interaction
formally the
by
some
construction
f.
model, system
at
as
interact
W of
non-occurrence
"looking
F a e L(Ha)
of the apparatus
In
time
interaction
apparatus
H,
apparatus
state
f is to be determined
ment
effect
the
again
occurrence
apparatus f
this
state
once and for all by the construc-
a certain
separate
the
the
Its
a
Before the measure-
not
the microsystem
during
unless
do
such
the apparatus
by
f on
in states
which
f, whereas Then
denoted
as follows.
prepared in
is
yes-no
in
operators
to which
apparatus
and is thus determined
arbitrarily.
here
the
state
apparatus
depend
assumed
the
and
a
f used
effect
apparatus
a state W e K(H) may then be described ment,
of
mechanically.
and
a.
performed
model
apparatus
subscript
measurement
representation
operations.
the
with
a new mathemati-
This
mechanical
same
applied, yes-no
that
previously,
operations.
quantum
measurement
Theorem
microsystem as defined
and in
§4,
apparatus and
are
together thus
form
a
binary
to be described
in
82
the
state
picture. means
space The
that
H = H @ Ha .
independent
the
It
is
convenient
preparation
incoming
state
of
of
the
here
to use
microsystem
composite
the
and
system
Dirac
apparatus
is uncorrela-
ted, W in = W @ W --
. a
Then the outgoing state is wOUt = S(W @ Wa)S*
with a unitary
"scattering"
all
the
effect
of
construction
operator
interaction,
of the apparatus
S on H which describes
the over-
m
and
f and
is
therefore
the kind
also
fixed
of microsystem
by
the
to which
it is applied.
(In
actual
experiments,
really
consist
single
microsystem.
is
put
again
microsystem ment.
the
not
under
one uses
state
change
time
different
of N
composite
of the apparatus,
Instead
into
does
is invariant at
of N copies
ensemble
Wa,
in the state W, after
This
formed
an
the
can
be
with
the completion
translations,
times
apparatus
combined
formalism,
will
each one combined
a single
and
systems
f only, which another
since
single
considered
with a
single
of a single measure-
however,
so that
not
as
the
theory
experiments
repetitions
of
per"the
same" experiment.)
Separate bed,
yes-no
both
before
measurements and
after
on microsystem the
and apparatus
are descri-
by operators
of the form
interaction,
G @ 1a and 1 @ G a with G E L(H) and G a ~ L(Ha) , respectively, whereas correlation effects are of the form G @ G a . Thus, in particular, the final the gives
"reading" particular the
probability
result
of
the apparatus
effect "yes"
1 @ Fa -
i.e.,
f is decribed
in
the
the
state
apparatus
as
W °ut. f
is
the measurement This
of
measurement
triggered
- with
83
f(W) = tr((l ~ Fa)W °ut) = tr((l 8 Fa)S(W 8 Wa)S*)
With W replaced nes
a linear
by an arbitrary
functional
f(T)
T e B(H)I,
on B(H) I. This
.
(5.1)
the last expression defifunctional
is continuous
with respect to the trace norm since, along with T and Wa, T ~ W a and S(T ~ Wa)S* are also trace class operators, and
IIS(T ~ Wa)S*ll 1 = l i t ¢~ Wall 1 -- IITII 1 ,
so that, as lJl ~ F a II = IIFa II < i, If(T) i < III ~ F a il IiS(T @ Wa)S*fl 1 < IITIII .
Therefore f(T) may be rewritten in the form
f(T) = tr(FT)
with
a
unique
operator
0 ~ f(W) = tr(FW) < i thus
belongs
probability
for
to L(H). tr(FW)
F e B(H). all
Since
by
the
Moreover,
W e K(H),
F
the apparatus microsystems
in
since
(5.1)
implies
satisfies
0 ~ F < I,
f is thus
triggered
the
(arbitrarily
and with
given)
state W, it indeed measures a uniquely determined effect F.
The state of the microsystems mined
by
measuring
interaction made.
Then
with
on
the
them
which leave the apparatus can be deterarbitrary
apparatus.
such measurements
are
effects
Assume to be
ments of the subsystem effects G @ 1a
first
G e L(H) that
described
no
after
their
selection
is
simply as measure-
in the state W °ut of the compo-
site system. These effects thus occur with the probabilities
w(G) = tr((G @ la)W °ut) = tr(GW)
,
where = Tr a -W °ut = Tr a (-S ( W ~ W a ) S * )
(5.2)
84
according to Ha; This
to (4.11).
arbitrary
Tr a denotes
the partial
trace with respect
is the reduction of the state o u t
i.e., shows
Here
that
the
initial
interaction
state
W
of
with
the
to the microsystem.
apparatus
the microsystem
into
transforms
a
final
an
state
given by (5.2).
The mapping
$ : W ÷ W
defined
by
(5.2)
may
be extended
immediately,
in the form ~T = Tra(S(T_ @ Wa)S*)
to arbitrary defines this
a completely
implies
until
T g B(H) 1.
a
that ~
little
operation,
Since
positive
one
an
however.
operation. As
a
easily
that
(5.3)
We
postpone
~
is
a
this
proof
non-selective
since (5.2) implies tr(~W) = trW = trW °ut = 1 for all W.
selective
the subensemble F. Assume
quite
expected,
The situation becomes more complicated perform
prove
(5.3)
linear mapping ~ of B(H) 1 into itself,
describes
later,
can
,
operation,
and
wants
of those microsystems
one has performed
if one uses the apparatus f to to determine
the state
which have triggered
the F measurement
of
the effect
N >> 1 times.
Then the
selected subensemble consists of N^ = N tr(FW) = N tr((l @ Fa)WOUt) _
(5.4) ^
microsystems,
according
semble
be
is
effects
to
G e L(H)
to
determined on
these
(5.1). by
Again
the
subsequent
microsystems.
state
W
of
measurements Such
this of
measurements
suben-
arbitrary give
the
result "yes" in ^
N+ = N tr((G ~ Fa)W °ut)
cases,
by
occurence site
defintion
of
the
correlation
of the subsystem effect G @ 1a
systems
selected according
effects
(5.5)
G @ F a . Indeed,
in the subensemble of
to the subsystem
effect
the
compo-
1 @ F a means
85
that both G @ 1a and I ~ F a occur together; and both measurements are performed in the state W out" The probability for the occurrence of the effect G in the selected
subensemble
of microsystems
is thus,
by
(5.4) and (5.5), N+ tr((G @ F )W °ut) w(G) =-~-- = a N tr((l @ Fa)W°ut) Due
to
the
cyclic
interchangeability
of
(5.6)
operators
under
the
trace,
the numerator of the last expression may be rewritten in the form I
1
tr((G @ 1a )(I @ Fa2 )-W °ut (I @ F f ) ) = tr(G.~W) with I
1
~W = Tr ((i @ Fa2)W°Ut(l ~ F 5)) a a 1
1
@ W a )S*(1 @ F a 2)) Tr a ((I @ Fa2)S(W -
(5.7)
'
according to (4.12).
Since the mapping Tr a is trace preserving, 1 I 2-wOUt(l tr(~W) = tr((l @ F a )_ ~ F 5)) a = tr((1 @ F a )W - °ut) coincides
with
the
denominator
of
the
last
(5.8)
expression
in
(5.6),
so
that we may rewrite this equation in the form ^
w(G) = tr(GW)
,
W = ~W/tr(~W)
;
of
microsystems
is
(5.9) ^
i.e.,
the
Moreover,
selected by (5.1),
subensemble
in
the
state
W.
(5.8) and the definition of F, tr(~W) also coinci-
des with the probability
tr(FW) of the measured effect F in the state
86
W,
which
tion
is
the transition
procedure.
(Thus,
probability
as
discussed
N/N
in
for
§2,
the performed
tr(~W) =
selec-
0 again
means
^
that the subensemble
is empty,
-
(5.9).)
be
defined
: W + ~W
via
defined
by
and therefore W need not - and can not
We
(5.7)
therefore
describes
expect
that
a selective
the
mapping
operation
in
the
sense of §2.
To show this, we first
extend
t h i s mapping
~ to B(H) 1 in an obvious
way, by defining 1
i
_ ~r = Tra((l ~ Fa2)S(T @ Wa)S*(I @ F a2))
(5.10)
for arbitrary T s B(H) I. For such T, the operator 1
1
(I @ Fa2)S(T_ @ Wa)S*(l @ Fa2)) belongs Ch.
to B(H) I, since B(H) 1 is a two-sided
1, or
Since
[4]),
Tr a
defines
It depends
maps a
linearly
on T, and
B(H) 1 into B(H) 1 and
positive
tr(~W) < I for
all
linear W,
as
(cf.
is positive
is linear
mapping
shown
ideal in B(H)
and
above.
Therefore
if T is.
positive,
~ : B(H) 1 ÷ B(H) I.
[3],
(5.10)
Moreover,
it only
remains
to
prove complete positivity of ~.
Take,
for
this
Hilbert
space
tensor
product
i.e.,
purpose, H,
and
of
the spaces
an arbitrary consider
Hilbert
the
spaces
(H @ H a ) @ H,
finite-dimensional product is
space
associative
(H @ H) @ Ha,
etc.,
or separable
H @ H @ Ha . and
The
commutative;
can all be identi-
fied with H @ ~H ~ H a in a "natural" and obvious way. Therefore operators like S @ 4, or T @ W a with T s B(H @ ~)I and W a e K(Ha), etc., can be considered as operating on H @ ~H ~ H a . Define now, for arbitrary T E B(H @ ~)I' 1
I
~T = --aTr((I ~ ~I ~ Fa2)(S @ ~)(T @ Wa)(S_ @ !)*(I 0 ~i @ Fa~))
with
Tr
--a
denoting
the
partial
trace
with
respect
to H a
,
(5.ii)
which
maps
87
B(H 0 H 0 Ha) 1 onto B(H ~ ~)i' and which therefore has to be dlstinguished
from the analogous
mapping Tr a : B(H ® Ha) 1 + B(H) I considered
before. Eq. (5.11) is of the same form as (5.10), with Ha, W a and F a unchanged,
but
H 0 H ~ Ha,
1 by
with
H
1 @ 4,
(which is also unitary).
replaced and
by
H 0 H
(and thus
Tr a by Tra) , T by T,
Therefore
H 0 H a by and S by S ~
(5.11) defines a positive
linear
mapping ~ of B(H ~ ~)I into itself. With T ~ B(H) 1 and T g B(H)I, we have
so that (5.11) implies 1
1
~(r 0 T) = Tra([(l 0 Fa2)S(T 0 Wa)S*(I ~ Fa2)] ~ !) 1
i
= (rra[(l ~ Fa2)S(T 0 Wa)S*(l O Fa2)]) ~
=
Here
we
have
~T
used
e T .
(5.10)
(5.12)
and
the
fact
that,
for
arbitrary
e B(H 0 Ha) 1 and T e B(H)I,
(5.13)
Tra( ~ 0 T) = (TraR) ~ T .
(The latter is easily proved, e.g., with the help of suitably generalized versions of Eq. (4.14).)
Taking
H
n-dimensional,
(5.12)
implies
that
~
coincides
with
the
mapping ~n used in §2 to define n-positivity of ~. As ~ is positive for
all n,
this
establishes
complete
positivity
infinite-dimensional H we obtain the (apparently)
of
~, whereas
for
stronger posltivity
property discussed before.
Because Eq. (5.3) results from (5.10) by substituting for F a the unit o p e r a t o r l a , t h e arguments o f t h e l a s t two p a r a g r a p h s a p p l y t o t h e
88
mapping Thus, tors
~
as
well;
indeed,
i.e.,
both
~
is also
linear
~ and $ are operations.
1 @ F aI/2 under
the partial
trace,
and
completely
By rearranging
one obtains
positive. the opera-
two equivalent
but
simpler versions, ~T = Tra((l
@ Fa)S(T @ W a )S*) -
(5.14)
'
or
~T = Tr a (S(T @ W a )S*(I @ F a )) , of Eq.
(5.10).
Such
rearrangements
are
possible
since,
for arbitrary
g B(H)I, X a ~ B(H a) and X ~ B(H),
tr(X.Tr a ((I @ Xa)T)) = tr((X @ la)(1 @ Xa)T) = tr((X @ la)r(l_ @ Xa)) = tr(X-rra(T(l_
@ Xa ))) '
and thus Tra((1
If the apparatus which
have
apparatus simply
@ Xa)T)_ = Tra(T(l_ @ Xa))
f is used
not
triggered
effect
Fa
by replacing
-
to select it
the
-
.
(5.15)
the subensemble
i.e.,
which
corresponding
have
not
operation
F a by F'a = 1 a - F a in (5.14).
of microsystems produced
~'
Then,
is
the
obtained
for arbitrary
T c B(H)I, ~T + ~'T = Tra[(l @ Fa)S(T_ @ Wa)S*_ + (i @ F~)S(T @ Wa)S* ]
=
Tr
a ( S-( T
@ Wa)S*)
= ~T
and thus, in particular,
tr(~W) + tr(~'W) = tr($W) = 1
for
all
W s K(H).
Therefore
~
and
~'
are
complementary
operations
89
(cf.
(2.37)),
and ~ is the non-selective
operation
~ + ~' associated
with ~ and ~', as expected from the construction of the model.
The
adjoint
~*
of
the mapping
~ is defined
implicitly
(cf.
(2.25),
(5.14) and (4.12)) by tr(~*X.T) = tr(X.~T) = tr((X @ ia )(I @ F a )S(T @ W a )S*) = tr((T @ W a )S*(X @ Fa )S) for
arbitrary
with
an
last
trace
X g B(H)
arbitrary
unit
and
vector
in (5.16) with
respectively,
and
T ~ B(H) I. f g H,
suitable
exploiting
(5.16)
After
inserting
evaluating
orthogonal
the
the
T = If >< f I first
and
the
bases in H and H @ Ha,
polarization
identity
(1.13),
we
obtain (f,%*Xg) = ~((f @ g~),(l @ W )S*(X @ Fa)S(g @ g~)) k a ' valid
(5.17)
for all f, g e H and an arbitrary orthogonal basis {gkl in H a .
(Compare
the
analogous
derivation
of
(4.14)
from
(4.12).)
Comparing
this with (4.14), we see that (5.17) can be rewritten formally as
(5.18)
~*X = Tra((l @ Wa)S*(X_ @ Fa)S)_
More
precisely,
when
defined
we have by
proved
(4.14),
(I @ Wa)S*(X @ Fa) ~
(which
(5.18)
with
holds
true
that
can need
this
not
the partical
be
extended
belong
extension.
trace mapping to
the
operators
to B(H ~ Ha)l) , and In
particular,
Tr a ,
the
that
effect
F = #*i corresponding to % is given explicitly by
F = Tra((l @ Wa)S*(I @ Fa)S).
Eqs.
(5.18)
and
(5.19)
are not very useful
in practice,
shall thus not be discussed further.
The results obtained so far are summarized and extended by
(5.19)
however,
and
9O
Theorem 2 (Second Representation Let
Ha
be
Hilbert
a
Theorem):
(finite-dimensional,
space, W a a statistical
Ha, a n d S a u n i t a r y
operator
separable
or
even
operator and F a an effect operator on
o n H @ Ha . T h e n
~T = Tra((l @ Fa)S(T @ Wa)S*)
with
T c B(H) 1 arbitrary,
the o p e r a t i o n
~'
defined
non-separable)
defines
an operation
,
(5.20)
~. With F'a = la - Fa'
by
~'T = Tra((l @ F~)S(T @ Wa)S*)
is
complementary
to
~,
and
the
non-selective
(5.21)
operation
~ = ~ + ~'
associated with ¢ and ¢' is given by
~T = Tra(S(T @ Wa)S*)
Vice versa,
given any two complementary
there
a Hilbert
a
exist
unitary
operator
represented also
by Eqs.
require,
(i.e.,
a
S
in
space Ha,
(5.20),
addition,
one-dimensional
operations
operators
on H @ Ha,
such
(5.21) that
projection
~, ~' and
(5.22),
is
~ and ~' on B(H)I,
W a c K(Ha) , F
that
and Ha
(5.22)
.
e L(Ha) , and
~ = ~ + ~' are
respectively.
separable,
operator),
Wa
and
a Fa
One may
pure a
state
decision
effect (i.e., a projection operator).-
Proof: that
The
first
part
of
the theorem has already
been proved.
(Note
the dimension
of H a in the above model was arbitrary.) To prove the second - and more interesting - part, we start from the representations (3.19) and (3.20) of ~ and ~', ~T =
as provided operators
~ Ak T ~ keK
by Theorem
Wa,
,
~'T =
I', and construct
Fa a n d S e x p l i c i t l y .
For
~ Ak T ~ keK'
,
(5.23)
the Hilbert space H a and the the
sake
of
definiteness,
the
91
index
set
J,
of
complementary without
which
subsets,
loss
of
the
is taken
generality
by
nal index set J was finite,
Take
for
Ha
a
sets
and
K'
occuring
to be J = {1,2 .... }. This setting
(5.13)
are
can be done
Ak = 0 for k > N if the
Hilbert
space
with
origi-
orthogonal
Then H ~ H a = H_ may be decomposed, =
in
J = {I...N}.
separable
{gia I i = 0,I,2...}.
K
basis
in the form
@ (H 8 g~) = @ H i , i)O i~O a
orthogonal
into
identified
subspaces
with,
H.
Hi = H ~ g i
Taking
the
isomorphic
subspaee
H
to,
~ H
and
apart,
therefore
we
may
also
O
write ^
H _
=
H
•
^
H
,
H
=
e
Hk
,
Hk
~ H
•
k)l In
the
following,
values
the
i = 0,1,2...
indices
i and k are
and k = 1,2...,
always
which
somewhat
assumed
to take
simplifies
the
the no-
tation.
Define,
in terms
of
the
operators
Ak,
k = 1,2...
entering
(5.23),
an
^
operator
A : H ÷ H = ekH k by Af = @ ~ f K k
(Note
that,
for
arbitrary
f e H,
.
Akf e H ~ Hk.)
Eq.
(3.18)
of
Theo-
rem I' then implies
(Af,Ag)
i.e.,
A
is
= ~ (Akf,Akg) k
isometric,
with
= ~ (f,~Akg) k
operator
norm
= (f,g)
IIAII = I.
;
Its
adjoint
^
A*
: H ÷ H, defined
by
(f,A*f) ^
= (Af,f)
^
for all f c H and f e H, is also bounded,
with
(5.24)
92
i(f,A*)J IIA*II = Sup
^ Ilfll
0(Af,f) = IIAII = I .
= Sup
Ilfll
Ilfll
II II
Isometry of A implies ^
A*A = 1 ,
AA* = E
^
with
some ^
projection
operator
(5.25)
^
E
on
H.
Indeed,
A*A : H ÷ H
and
^
_AA* : H ÷ H are obviously self-adjoint,
and A*A = 1 since
(f,_A*Ag) = (Af,Ag) = (f,g)
;
thus ~2
Explicitly,
^ = AA*AA* = AA* = E .
A* is given by
A*
: • fk k
÷
(5.26)
~ ~fk k
^
for arbitrary @kfk e H (i.e., fk ~ H, ~k;Ifkli2 < ®), since then
(A*(O fk),f) = (~ ~ f k , f ) k k
= ~ (fk,Akf) k
= (0 fk' • Akf) = (0 fk' ~f) k k k i.e.,
;
(5.26) is satisfied. ^
Vectors ! e H = H • H may be represented
^
with
f e H and
in matrix notation as
^
f e H.
In this notation,
be written as operator matrices,
=
Xl
X2)
X3
X4
bounded
operators
X on H can
93
with bounded operators ^
XI
: H
÷
H ,
so that, according
X2
^
: H
÷
H ,
X3 : H
+
^
H ,
X4 : H
^
+
H ,
to "matrix multiplication",
(xif+x2 ) x f= ^
X3f + X4f The adjoint of X is easily seen to be
x(X while
operator
products
to
be
calculated
ordering
of
the
tion",
preserving
ments",
of the operator matrices.
We
define,
now
in
the
are
this
matrix
S = -
Obviously S* = S; moreover,
notation,
A
the operator
-S
multiplica"matrix
ele-
on -H - - H
@ Ha
. ~
-
(5.27)
AA*
S is also unitary:
s.s=ss,=s2= ( . . . . .
"matrix
(non-commuting)
(0
by
by
A*(~-~*)
A*A (I
-
AA*)A
./j~* +
(i" -
)
_A/j*) 2
(i 0) 0 Here ^
we
have ^
used
the
~
=i_.
following
facts:
By
(5.25),
^
1 - AA* = 1 - E is a projection operator,
A*(I-~*)
=A*-A*~*
so that
= A * - A * =0 ,
( I - ~*)A= A-~*_A= A-_A= o ,
A*A = i,
and
94
and AA* +
- .AA*) 2 . = .AA*. + . i^ - A A * .
(1
--1
= I^ .
a
For W a we take the projection operator ont 9 go' a
a
Wa -- I g ° >< go I .
Then we obtain,
(5.28)
for arbitrary T e B(H)I and f e H,
(5.29)
(T 0 Wa)(f O g~.) = ~jo(Tf @ goa) .
a
Due
to
the
may
also
isomorphism
be
written
H @ H a -- eiHi,
as @i~ji f.
In
H i -- H,
the
the vector
above
matrix
fj = f 8 gj
notation,
this
means f = -o
, 0
f. = -J
for j > 0 , jk f
so that (5.29) takes the form
(T @ Wa)f ° =
(Tf)
(T @ Wa)f j --
(0)
Since
vectors
of the
for j > 0 .
0
0
form -j f. span _H, this means
that
T @ W a has
the
matrix representation
T wa (T0 0°I Define
now,
for
an
arbitrary
(finite
or
(5.30)
infinite)
subset
^
J = {1,2...},
a projection operator EK on H by ifkeK E (e fk ) = • gk ' k k
gk = $ fk L 0
ifk~<
and two projection operators,
m Ka =
~ I g k >< gka I , kgK
Ea O,K
=
I goa >< goa I
+
E aK
K
of
95
on H a . Then,
by arguments
similar to the ones leading to (5.30), one
easily verifies the relations
(0
1 @ Ea
--
,
0 A
Ea
straightforward
1 ~
(i o)
=
O,K
0
K
E
(5.31)
"
K
calculation
with
(5.27),
(5.30)
and
(5.31)
then
yields dE. (i @ E~)S(T @ Wa)S(I @ E~)
a
=
~-
(i @ Ea,K ) s ( T-o
@ Wa)S(I- @ Eo,K)
(0 01 ^
0
(oo)
^
K--
--
(5.32)
^
E ATA*E
u~.
0
B
K
Since T e B(H)I , we have B e B(H) 1 , and thus TraB e B(H) 1 ial
trace
with
the
can be calculated particular
basis
by
(a suitable
{g ~}
in
Ha
analogue
used
The part-
of) Eq.
before;
for
(4.14)
arbitrary
f, g E H we then get
(f, TraB g) =
~ ((f @ g~), B(g ~ g~)) .
(5.33)
j~o As above, we have in matrix notation
(g) 0
(0) ~k 6 jkg
and similarly with f instead of g. Thus (5.32) and (5.33) lead to (f, TraB g) =
= By definition,
E
j ~ ~. Therefore
~ ((O 6jkf), B(O 6jkg)) j>0 k k ~ (A*E (~ TA*E (O ~jkg)) j>O - ~ k ~Jkf)' - K k
reproduces the operators
•
@ ~ j k f if j g K, and annihilates E
it if
may be dropped in the last expres-
96
sion,
if the
sum
over
j is restricted
to the
index
set K. Eq.
(5.26)
implies A*(@ ~jk f) = A@f -
so that,
k
J
finally, (f, TraB g ) =
~ (A~f,TA~g)=
j~<
~ (f,AjTA~g)
;
j~K
i.e., Tr B = ~ A. T A@ . ajCK J ] (Note
that,
as
shown
in
§3,
the
the case where K is infinite,
On
the
other
hand,
by
the
sum
in
(5.34)
(5.34)
is
convergent
also
in
since ~kA~Ak = I.)
definition
(5.32)
of
B,
and
since
S* = S,
we also have Tr a-B = Tra((l
@ E~)S(T @ W a )S*) -
= Tr ((I @ E a K)S(T @ Wa)S*) O,
a
Here or
we
have
also
1 @ Ea
under
used
Eq.
(5.15)
the
partial
.
(5.35)
--
to bring
trace
to
the
the
last
left,
operator where
it
1 @ Ea can
be
O,K
omitted
Ea
since
and
K
(5.34) tions
and
(5.20)
(5.35),
Ea
O,K
are
we now s e e
and (5.21)
projection that
operators.
we o b t a i n
the
of ~ and ~' by taking
Fa = EKa
'
Comparing
required
Eqs.
representa-
either
F'a = E a o,K'
'
or
F (By
(5.35),
it
does
K u K' = {1,2,...},
a
= Ea o,K '
not
matter
F' = E~ a ' " which
of
these
we
choose.)
we have in both cases
a a a a a a Fa + F'a -- ~ i gk >< gk i + I go >< go I + ~ I gk >< gkl ks K kcK ' a a = ~ I g i >< g i I = 1 a . i>0
Since
97
As announced projection
in
the
theorem,
operators,
and
the
Wa
as
operators defined
Fa
by
constructed
(5.28)
here
describes
are
a
pure
could
also
state.
We
finally
remark
be carried if
the
that
the
above
contruction,
through with a Hilbert
set
of
operators
Ak,
obviously,
space H a of finite dimension
k E J
entering
(5.23)
is
N + 1,
finite,
i.e.,
J = {I...N}.
The
theorem
just
proved
allows
some
conclusions,
which
shall
be dis-
cussed now.
We have
shown,
apparatus
and
measurement tions
to
its
of
~ and
ding
first,
the
effect
second
~ and
a suitable
apparatus
choice
of
Fa
the
of
This, red
part
~'
does
and
not
could
complementary
opera-
Conversely,
accor-
the
two
measurement.
theorem,
initial
prove
be
any
at least
already
interacts
formally,
that
such an apparatus
reproduces
possible
the very
S
general
this,
2 provides
F e L(H) insists by
not
only
describe in
possible
"conventional"
postulating
projection
projection
that
operator,
the the
operators
yes-no quantum
"pointer" effect
E,
support but
for
rather Indeed,
mechanics
the
effect
F a is always
F measured
by
such
our
all
measurements. for
In
model
class
Besides
that
the requi-
concrete
in a much more abstract way in §2.
tion
H 8 Ha .
it is, neverthe-
as introduced
additional
effect
laboratory.
rations,
Theorem
on
-
with
the
of quantum mechanics
the simplest
with the
"pointer"
operator
in
by
by an appropriate
W a and
constructed
the
"in principle",
which
state
describes
of complementary
scattering
actually
that
pair
the
consistency
satisfactory
measurements
and
way - i.e.,
of
of an effect
indeed
"realized",
Ha,
model
microsystems
this
the
be
space
view of the internal
yes-no
of
mechanical
of the type considered,
state
interaction
quite
with
could
apparatus,
clearly,
less,
with
in an appropriate the
quantum
F e L(H)
associated
operations
microsystem
our
interaction
an
~'
that
of ope-
assump-
operators
even
model
of
if
one
apparatus
described
by a
an apparatus
need
98
not
be a projection
F E L(H)
can also
operator.
be measured
apparatus,
according
operations
~ performable
ly
general,
e.g., not
the be
so
F
condition
and
whether
or
states
not
is
and, (2.9)
-
e.g.,
cular
not
that
time
(2.8)
by
During
system,
and
formalism,
this thus
quoted
evolution
which
is
picture
applies
again
- leads
of a
above the
closed
system,
We have
shown,
very
measurements
One
and
argues,
for
described,
in the SchrS-
pure
states
-
(2.8)
(as
states,
into
whe-
mixtures.
Not only the parti-
pair of complementary
from
the
certain
of
the
interaction interval
choice
the "reading" the
(isolated)
system and
is
indeed, apparatus described
reduction
moreover,
as
of
the
time.
of the appara-
microsystem
systems
of
- or even the
can not be described
to a state W °ut whose
the state ~W.
to
into
every
a
is misleading;
composite
look
the non-selectl-
interval
to closed
of
In these models,
describing
time evolution
ef-
quantum mechani-
is independent
interaction
of
by a suitable
during
Fa
not
mechanics
pure
in fact
apparatus
effect
its
which
t h e argument
just
the
can
apparatus
formulae
is unfounded.
results
~
would
mechanics.
most
process.
kind)
initial
due
states
simply
(5.22),
- of
but
(as,
independent
for the state vectors pure
~
classes
are
reduction"
can be reproduced
the
Wa
completeon
first
the
as
changes
quantum
reasoning
(2.9),
~ = ~ + ~'
(Note
tus.)
and
in
the
the
nature of the apparatus.)
quantum
transforms
of the measuring
with
by
transforms
this
~ and ~'
that,
states
packet
equation
(2.9)
that
microsystem
existence
"wave
of
the model
pure
operator
Moreover,
restrictions
admitted
state
evolution
always
2 shows
operation
time
be
operation
cal model
by
2.
are also
(Likewise,
are
to
that
explained
operations
ve
claimed
can
cases
either.
states
the
the
apparatuses
in view of the macroscopic
picture)
Theorem
way
pure
Theorem
additional
~ described
by the SchrSdinger
dinger reas
this
particular,
instance,
of
every effect
by such a "conventional"
for measurements
restriction
frequently in
part
by such model
~*F = F
in
a
last
conceivable
only
W a . But
"in principle"
the
operations
natural anyway,
It
to
that
justified
fects
On the contrary,
is a n
open
by the usual
only.
Therefore
shown
here,
plus here
microsystem in
the
-
Dirac
to the microsystem
that the selective
the
is
opera-
99
tions
# and #' can be also understood
in terms of conventional
quan-
^
tum mechanics.
Since
produced
are
by
effect Fa, composite a
~
always
in,
preceded
e.g.,
by
the
the
state
reading
W = #W/tr(#W)
of
the
apparatus
one is in fact dealing with correlation measurements system.
convenient
ments.
measurements
This
In our model,
"shorthand" description
apparatus variables mapping
Tr a . We
belief,
state
description
is obtained,
of as
~ and #'
such in the
just provide
correlation
measure-
case
when
of ~,
the
are eliminated with the help of the partial trace
may
thus
changes
well understood,
the operations
on a
of
conclude the
that,
type
(2.8)
if quantum mechanics
contrary and
to
(2.9)
a wide-spread
can be perfectly
is assumed to be valid also for
measuring instruments.
Our
model
complete
also
positivity.
completely
positive,
scribed by Eq. for
illustrates
all
in a simple way
An operation if
(5.10) and
(5.11) are positive
(5.11), an
~ of B(H @ ~)I
for arbitrary
Hilbert
spaces
one immediately realizes
operation
preted
as
acting
on a composite microsystem which
ing subsystems
performed
physical
~ as given by Eq.
the mappings
finite-dimensional)
the
by
meaning
(5.10) into
above
itself
is de-
(and, in particular,
H.
Comparing
now
Eqs.
that ~ may also be inter-
the model
apparatus
consists
f,
but
now
of two noninteract-
I and II with state spaces H and H, respectively,
is thus described
of
in the state space H @ H. The model apparatus
and f is
the same in both cases,
since the state space H a , the initial apparatus state W a and the "pointer" effect F a are identical in Eqs. (5.10) and (5.11). Moreover, the unitary operator in (5.11) describing the "scattering"
between
the composite microsystem
I + II and the appara-
tus f is given by S @ ~. Obviously this means that, first, the apparatus
interacts
fected,
and second,
apparatus since
only with subsystem I while
the
is
independent
resulting
same operator
that the interaction of whether
"scattering"
is
leaving between
or not
S on H @ H a . This interpretation
of our quantum mechanical
model
is therefore
subsystem
subsystem
described
--
subsystem II unaf-
in
II is present,
both
of Eq.
I and the
cases
by
the
(5.11) in terms
in full accordance with
the physical meaning of the mappings ~, as discussed in detail in §2.
100
In spite
of its considerable
overestimate reasons
the
to
doubt
appropriate
This is a highly
nor
require
really
latter
should
usually
easily
though into
called
quantum
attempts
mechanics,
have have
the
been
resulting
our
model
would
interest. trons. an
be
If
now
space
totally
trons.
still
be
For instance,
electron,
state
satisfactory
mechanics
In
measuring
then for
the
combined
antisymmetric view
of
process
contains
all
should
product system,
with
are
validity the
also
of quantum
system
described
in
rather than a macroscopic
to
quantum
description
of Theorem
true
are relatiAl-
such
ideas
yet be consi-
macroscopic
cases
of
sy-
practical
contains
too,
or
elec-
is itself
of
the
model
as a model only,
elecof
the
rather
of actual measurements.
2 which
space f.
set of
for macrosystems,
mechanical
for macrosystems.
apparatus
of
permutations
be considered
state
in
The
properties.
can not
electrons,
really
the
system.
which
instrument
our
mechanics
majori-
observed
incorporate
in many
this,
applications
subsystems.
H @ H a is not the appropriate since all state vectors have to
respect
than as a complete and realistic
There
to
literally
oversimplified
tensor
the
"classical"
every real measuring
the
in-
of the universe),
of
description
were
the microsystem
measuring
atomic
ones,
theories
in spite of some partial successes.
quantum
the
body as a compo-
the size
made
stems,
if
fully
is
in terms of a much smaller
suitable
good
since their measure-
behaviour
to
even
a
at all,
dered
But
yield
1024
the "macroscopic"
and
e.g.,
are
form
system are neither
actual
be describable
There
present
like,
exceeding
the
one should not
since the overwhelming
a complex
for
its
about
not even observable
measurable
numerous
least)
"instruments"
rather
in
a macroscopic
description,
such
significant
observables, vely
of
however,
the model.
systems
describes (at
redundant
(and perhaps
ment might
of
of
mechanics
o f macroscopic
"observables"
practice
quantum
consisting
value,
significance
Quantum mechanics
system
ty of
that
theory
struments. site
physical
heuristic
Ha
do not presuppose
the
In such applications, is
also
a microsystem,
i01
Denote
by s and a the microsystems
tively,
and
assume
tering
system
that,
when
put
together,
they form a binary
scat-
s + a. If N pairs s + a are prepared in an uncorrelated state W @ W a = W _ in , their state after the scattering is
incoming
wOUt = S win S,
with
above,
the N systems
and thus
Keeping
the
initial
procedure
yields
system
as
s,
systems
of
state
operator
Eq.
of
into
two
varying
2.
measurement
F a performed then
"yes" at
W,
as
this
for
Moreover,
complementary
to the outcomes
One
H @ Ha,
~ : W ÷ W
Theorem
respect
scattering.
on
in the state W = Tr aW °ut.
operation
(5.22)
separated
the
S
of system a fixed while
nonselective
by
with
after
scattering
s are finally
Wa
a
be
a yes-no
s + a
unitary
described
s can also
tively,
a
again
they are selected
pair
with state spaces H and Ha, respec-
the
the N
ensembles,
or "no",
subsystem
concludes
if
respec-
a of each
as
above
that
W = ~W/tr(~W)
and
^
these
two
subensembles
W' = ~'W/tr(~'W), Theorem
2,
and
with
are
~
consist
and of
in ~'
the
given
states by Eqs.
N+ = tr(~W).N
and
(5.20)
and
(5.21)
N_ = tr(#'W).N
of
systems,
respectively.
An
"indirect"
measurement
complementary ding
effect
strument by Wa, but
F = ~*i
w a for systems
insensitive
system
apply
composite perform
or
operations
F will
regardless
restriction
are to
s. The
combination
also
may
be
of
a
fa
the
~' not
or not
projection and may
the
and
of the corresponof a preparing
together
in-
be
to
and
a,
Then,
effect
F,
~ = ~ + ~'
be described
as a single
system s in the follow-
let it
experiment.
measures ~,
to a single
apparatus
two
the effect F a on system a
considered
Wa,
yields
a single system a prepared
fa measuring
from
not uncommon
is unnatural,
of the theory.
s,
the
in general
of whether
measurements
ments
f
therefore,
a in the state Wa,
system
"no"
apparatus the
system
f, to be applied
then
type,
~ and ~' and a measurement
system
the
"yes"
this
apparatus
release
s,
outcome
a
to
apparatus
way:
effect
on
and an effect
effect ing
operations
of
at
as
scattered read shown
and
may
system
on
at
the
fa
the
above, be s.
by a projection
used Again
this to the
operator,
this
is true for F a . Since such indirect in practice, this shows once more that
operators in fact
as
lead
describing to internal
yes-no
measure-
inconsistencies
102
It has even been argued
([13], Ch.
II) that typical measuring instru-
ments always contain a suitably prepared microsystem a (a "trigger"), which
first
eventually
interacts
triggers
with
could
describe
observed
some observable
pic part of the apparatus we
the
the
s,
and
afterwards
change on the remaining macrosco-
(the "amplifier").
interaction
system
between
Taking this for granted, a and
s as
a
scattering
process, uring
and consider the "amplifier" as an effect apparatus f a measa certain effect F a on the "trigger" a, thereby arriving at
exactly the situation considered above. Whether or not such a description of the quantum mechanical measuring process listic
and
really
helpful
discussed here, however.
for
a
deeper
is sufficiently
understanding,
rea-
shall not
be
103
§6 Coexistent
A
set
of
effects,
coexistent applying
are
either
ratus
apparatus here
"yes"
or
in
tr(FW)N
combination ment
one
"no",
and
of effect
of all effects
The
each
which
-
effect respond
to the effect
apparatus
F E C when
c may
f, which
with
set
of
together
by
as having
whose
the
i.e.,
outputs
appropriate if the appa-
in a state W,
F e C gives thus
be
perform
the combined
a
Such an appara-
F e C,
to N >> 1 systems
apparatuses
or:
be visualized
of microsystems;
corresponding cases.
c - may
for
successively
channel
coexistent
F e C can be measured
label
to ensembles
c is applied
"yes"
called
to single microsystems.
by the
channels",
frequencies
the output
is
- if all effects
a suitable
"output
relative
C = L(H),
effects
tus - abbreviated several
Effects and Observables
then
the output
considered
as a
the joint measure-
apparatus
c is applied
to
a single microsystem.
In
conventional
operators)
quantum
are
are usually a
considered,
with
out
to be equivalent
ries, rable. trary
slightly
features
that
there
We want
only
decision
coexistent (Ludwig
meaning;
of
quantum
are sets
theory,
now
sets
of
(projection
decision
[2] defines
effects
"commensurabili-
however,
this
It is one of the most as
of decision
effects
finally,
to "coexistence".)
The notion
compared
to
classical
turns charactheo-
effects which are not commensu-
the corresponding
of "coexistence"
problem
as defined
above
for arbiis due to
[2J.
Commensurability verbis,
narrower
to investigate
effects.
Ludwig
and
called commensurable.
ty"
teristic
mechanics
or
at
in quantum least
tentatively
measurability.
Nothing
definition
coexistence;
of
stem with all parts of the different
mechanics
of this
-
sort
neither
as
channels
considered
synonymous
is implied, the
of the "composite"
output
is often
need
with
apparatus be
"simultaneous"
however,
interactions
- expressis
of
by the above the microsy-
c, nor the responses
"simultaneous"
in any sense.
104
This
may
be
apparatuses effects
F
(Note
that
the
interacts whole
apparatus
simple
f acts
is
If
and
earlier
occupy
can
put together,
and applied
single
microsystems.
The
single
apparatus
channels",
this
c.
Ig
f and
do not be
g,
apparatus
c
Ig.)
and
that
the
with
the
in the speciis relevant
the microsystem we assume
Assume
also
that
that
the
the apparatuses
f and g
Then both apparatuses
f and g
successively
combination
which
respectively;
than
effect
with the appara-
are included
during
region in space.
Having,
measurement
f and g. What actually
non-destructively,
the same
the
two
would measure
than the measurement
apparatuses
If
that
of application
the apparatuses
Consider
separately,
Assume
earlier
intervals
interval
example.
when applied
the times
time
with
a
respectively.
of the effect
are
the
G,
f is performed
fication here
by
f and g which, and
apparatus tus g.
illustrated
f + g
- first may
by
construction,
is
expected
to
be
two
f, than g - to considered
different
measure
two
as
a
"output
coexistent
effects F 1 and F 2.
These te -
an
the
F 1 and F 2 are easily determined.
probabilities
i.e.,
for
of the parts
arbitrary
stems f,
effects
in this
triggering
nonselective state
state
the
W.
When these
it
tr(FW)N
in
operation
c
is
applied
systems
first
cases.
performed
therefore
After
by
f,
cases.
lities
and
question
the
are
tr(FW)
two "output
successively interact this,
to
with
with
the
Since,
channels"
c - by microsystems
the N systems
triggering
tr(GW)N = tr(G.~W)N = tr(~*G.W)N in
of
f and g of the apparatus
state,
W = ~W,
triggering
To do this, we calcula-
~
N >> 1 sy-
the apparatus denoting
are
tr(~*G.W),
the
the
in the
apparatus
therefore,
in
g
new in
the probabi-
effects
measured
together by c are
F1 = F ,
According obviously,
to our
definition,
F 2 = $*G •
these
they are not measured
two effects
(6.1)
are
"simultaneously".
coexistent
- but,
105
If,
in particular,
both F and G are d e c i s i o n
F =E
and the a p p a r a t u s
f performs
,
G=E
an "ideal"
effects,
,
measurement
(cf.
(2.9)),
~W = EWE + E'WE'
w i t h E' = i - E, then
(6.1)
reads
F1 = E ,
In
this
only
if
[E',E]
case E
F2
is
also
a
and
E
commute.
F 2 = EEE + E'EE'
decision
effect
Namely,
if
•
(6.2)
(and
equal
[E,E]
= 0,
E)
we
if
also
and have
= 0, and thus
F 2 = E2E + E'2E = (m + E')E = E
Vice
to
versa,
if
H = F2E = EEE commuting
F2
is
since,
a
by
projection
projection (6.2),
operators
operator,
F 2 commutes is
again
the
with a
.
E,
same and
projection
is
true
the p r o d u c t operator.
for of The
operator A = EE - EEE then s a t i s f i e s A * A = EEE - E E E E E ~
which
finally
= H - H2 = 0 ,
implies 0 = A = A* = A* - A = [E,E]
As
a
"physical"
volumes ly, stic
V 1 and
therefore functions
example, V 2 and measuring
consider
operating -
when
two at
counters,
times
applied
t I and
.
occupying t 2 > tl,
separately
-
the
spatial
respective-
the c h a r a c t e r i -
106
E = XvIX(tl))
of
the
position
operators When
at
applied
operators
different
times,
after
first
ures
an
effect
that
the
counters
not
be
X(t)
the
F2
which
satisfied
by
of
counters
involved,
and
most
at
t = tI
thus
-
t = t 2.
But
position
do not
commute.
one,
therefore,
the second
not
a
effect.
decision
counters,
if
likely
and
also E and E,
instantaneous
actual
scription
E = X.v2(X(to))
and
is
perform
,
and
possible
would
at
show
A
all
that
(Our
"ideal"
however.
assumption
measurements more
-
even
counter meas-
will
realistic
would
be
de-
much
more
counter
does
commute,
they
a single
not measure a decision effect.)
On
the
can
other
hand,
be measured
of
the
type
decision
if two projection
together-
Indeed,
and
sufficient
for
quantum
mechanics.
A
ry
at least
considered,
effects.
operators
and
"in principle"
therefore
should
commutativity the
and
known
of
rigorous
- by an apparatus
describe
is well
commensurability
complete
E and E
commensurable
to be neccessa-
decision
proof
of
effects
this
in
criterion
will be p r e s e n t e d below.
In order
to obtain
the
case
of
if
there
two
considered brevity,
By
to
output versa;
The
two as
F I and
apparatus
outputs
results
some yield
-
c
both
F 2.
criterion,
They
are
yielding, either
of measurements
wiring
an output
if
of
electronics, yes-no
which
is
2
and
only
are if
"yes"; at
realization
and
least of
an
one
or
"no" F 2.
besides output is
output the
if and to
a
- which For
the
only
single can
be
sake
of
i and 2 here.
the apparatus
if
of
applied
F 1 and
outputs
"yes"
we first consider
coexistent
when
"yes"
1 I% 2 = "i and 2" which
I and
technical
and
additional
I' = "not I"
outputs "yes"
an
coexistence
these two outputs are simply called
adding
fied
effects,
exists
microsystem,
a general
1
"yes"
c can be modi-
1 and is
2;
"no",
e.g., and
an vice
if and only if both
1 V 2 = "I or 2" which outputs
the corresponding
i and
new output
2 is
is
"yes".
channels
in
107
terms
of
well
the
known
to
continued, more
already
existing
every
thereby
complicated
channels
experimentalist. leading
when
also
expressed
for This
to
new
in
terms
the
outputs
procedure
outputs of
can
which
the
1
and
be
look
2
is
further somewhat
original
outputs
1
and 2.
As
is
also
operations
well
known
with
outputs
"or"
satisfy
rules
of a Boolean
usual
symbols
y,
etc.
(and
the
denoting
from
characterized
rules
of
by
ordinary
algebra.
' for
obvious
arbitrary
for
the
and
the
for
the
"and"
and
calculational
already
v
the most
examples),
"not",
i.e.,
we have
"and"
outputs,
given
words
logic,
(Therefore
"not" , A
the
introduced
"or".)
important
With
the
a,
B,
calculational
rules are : ~ A ~ = ~ ,
~ A B = B A ~ ,
({l A 8 ) A y = {~ A (IB A y )
and analogous
rules with
A
v
for
; (6.3)
(=')'
= =
,
(=
~ A ~' = ~ ,
vB)'
=
~ V~'
~' ^
B'
= I ,
,
(= AB)'
=
~' V
S'
,
~ A I = ~ V ~ = ~ ;
and
-~
(the puts:
distributive I,
which
Corresponding
^.(BvY)
= (~A
B) v (~ ^ ~ )
v(BAy)
= (~v
B) A ( ~ V ~ )
laws). is
In
(6.3)
always
output
there
"yes",
channels
and
are
occur
"l.
/ the
~ = I',
also
,
two
which
easily
(6.4)
"trivial"
is
always
incorporated
out"no".
in
the
given apparatus.
Starting
from
the original
well-known
technical
thus
lead
to
This
"enlargement"
to a "natural" in each single
manipulations
modified
end,
apparatus
of
apparatuses the
apparatus,
however,
experiment,
leading
yields
c with as
only
two output
symbolized
with
additional
when finally
pursued
by
' , A output
far
and
v
channels.
enough,
to an apparatus
the 16 outputs
channels,
comes
b which,
108
I,
1 A 2,
2,
I' ,
1 A 2' ,
(i A
I v 2',
using
that
the
any
further
16 outputs ment
calculational
of
does not enlarge
the
duplicate
apparatus
already
realized (I A
1 A
((I A
Being
V
, the
As
all
set
1 and
the outputs
16
with
~ c B are 2, B is
in fact
look
different
ting
the
channels from
other
when
I
Therefore
we treat
here
cases
it may
these
outputs
every
single
F 2 = 1 - F = F'
coincide
experiment. are
and
(i.e., For
coexistent
to these
so that a further
enlarge-
new
outputs,
2
to
the
but merely
either
all
every
(e.g.,
"yes"
the
F E L(H):
algebra.
or
effects
to the
by them.
technically
themselves
and
containing
(6.5)
nevertheless,
instance, for
algebra
outputs
suitable
happen,
' , A
operations
generated
realized with
I = 1',
is a Boolean
Boolean
the
2))
2',
these
algebra
to
(1' A
operations
to (6.9)
the outputs
are
(I A
= I'A
by applying
the Boolean
outputs
In particular
checks
v
2')) V
the smallest
original
rent.
easily
' , 2k and
(2 V 2 ' )
(6.5)
corresponding
each
one
2') V ~ = 1 A
respect
obtained
I and 2 - i.e.,
output
devices.
closed
(6.8)
(6.4)
yield
(I A
2') = I' A
B of the 16 outputs
outputs
outputs
The
thus
(6.7)
2',
= (I A
etc.)
2),
(For instance,
2)' = I' V
2) V ( I ' A
and
list,
2') V (I' /k 2)) = (I A
(I' A
A
(6.9)
really
ones.
(6.6)
I' V 2 ' ,
operations
the given not
2,
(6.3)
of the
would
2 ',
~.
rules
application
1'A
2') v ( l '
I' V
I,
By
(6.5)
1' 2k 2,
(I A 2) v (l'/k 2'),
I v 2,
2' ,
by
to
(6.9)
connec-
electronic)
also as diffethat all
some
of
"no")
in
F1 = F
If an effect
and appa-
109
ratus of
f with
the
second
apparatus
output
channel
an
above,
It is obvious, always
which
less
than
dered,
16
the
output
also
treat the
~
and
single
experiment,
(E.g.,
the
Such
four
a v 8 when
shall, = (~V
advantage
of
want
however,
an
not
(~' v
output,
More
generally,
al...
an
to
use
with
(6.9)
an
is
~i''" an.
generalization
b with
just to
as
consi-
(6.9)
the
two
apparatus
(a 2 V
...V
"universal"
said
never to
each
(~2v
Boolean
and
narrower
meaning
a ~ 8
8)
the
form
in
aI v
each
other,
the
disjoint
n = 2
to
each
shall
be
of
for
with
above
an) = (a I A
a2) V
also
union
n > 2
... V
... V an ) = al ~" (a 2 V
(a 1 A
... v
In
of
is
the
natural
an) = ~ ,
a n)
, etc.
this
written (We
~ or 8"
~A
under
a
other.
8 ~ ~. (6.7).)
a 2 V ... V a n, is
in
"or".
"either
cases
listed
"yes"
other.)
a or 8",
is
from
exclude
exclude
8 = 2,
called
both
and thus aI V
Our
different
by (6.4), aI A
to
"tri-
c.)
formally
same
are
notation
of
an,
This
are
excluding and
B--~
8') V (~'/~
a I ~" a 2 ~> ... ~
the
alge-
cases.
this
output
pairwise
the
only
b
identify
"output"
(6.5)
that
to
this
a = 1 and
so
apparatus to
example
as
channels.
apparatus
list
(6.5)
"either
8') = ( a A
the
mutually
stress
an
the
to
a A
(6.6) means
(In
~,
tempted
by
and
enlarged,
output
Boolean
added
therefore
outputs
actually we
8) A
and
at smaller
yielding
8 satisfying
be
be
B also in such exceptional
outputs
~ 8
outputs
F 1 = F,
I' and 2 of this
reduces
to
f, so that f
for
16 different
might
I and
corresponding
then be further
channels.
I',
A
1 = a
technically
I' = 2
have
all
one
realized
i,
outputs,
the outputs
output
outputs,
however,
case,
be
F'.
a' = "not a"
to the apparatus
b with
arriving
the output
measure
c may
cases,
thereby
channels
to
algebra
such
identification
vial"
decision
In
different
different
two
apparatus
that
to
be added
yielding
however,
may
four
Two
c
F, then
used
thus
This
outputs,
bra,
may
be
to an apparatus
coincide.
coinciding
a measures
may
apparatus
for F 2 = F'.
described
has,
output
same
becomes 2 = a'
yes-no
with
written effects since,
110
When
an
single red
apparatus
b
mlcrosystem,
as
the
result
of
the
type
each
one
of
of
a
measuring
a certain
apparatus
b thus measures
form F
a certain
subset
@ FS, B contains
at most
above
-
correspond
puts
do
effects
B of effects nal
in
may nevertheless
outputs
i and
effects 2,
and
tus
realize
said
to
joint
measurement
to
the
result
a
of
but possibly
in the example But
even
experiment,
The
F , ~ e B, which
not necessarily
effect.
the
imply fewer.
discussed
if two outcorresponding
as we shall see later on.) The set and it contains
corresponding
Therefore
coexistent
is
same
F2
pletion of the original b
- as
single
respectively.
effects
effects,
by definition,
F1
i.e.,
As ~ @ 8 does
be equal,
applied
~ e B can be conside-
-
these
outputs
every
is coexistent,
coexistent
all
to the
been
- on the given mlcrosystem.
16 different
coinciding
coincide
16 outputs
e L(H)
B of L(H).
two
not
F
has
measurement
together
(In particular, clearly
its
yes-no
effect
considered
to
the
particular
we call B a coexistent
set of effects this
the origi-
the appara-
{FI,F2};
coexistent
com-
completion
B
of
{FI,F2}.
Although
the
visualized an b
- and therefore
"enlarged" is
not
and
~
really
are
vial"
apparatus
outputs
calculated (For
from
the
instance,
if
microsystem
yields
outputs
2
IA
(i A 2) v (I' A nal F
apparatus
the actual
and 2') c
for
outputs
i' A
for
2'
in
in (6.7)
already
e B
purpose.
under
I and 2 of of
this
output
1
(6.6)
are
is also
measures,
and
"no", at
The
of
this
easily
apparatus
"trivial" and
(6.5)
all
I
"nontribe
the original
apparatus
c.
apparatus
to
"no"
etc.)
to
outputs
can
"no",
least
is most
here - in terms of such
construction
above
application
"yes"
F
to any measurement,
listed
two the
this
prior
as
effects
(6.8),
c
for
a
output
therefore Therefore
implicitly,
single 2,
the
the
output
the origiall
effects
e B.
The
correspondence
mapping F
b,
fixed
~ e B,
all
has been discussed
necessary
already
of
are
F : ~ ÷ F immediately
between
outputs
~ ~ B and
of B onto B c L(H). obvious
from
effects
The properties
its physical
meaning.
F
e B defines
a
of this mapping First,
obvious-
IIi
ly,
the
trivial
outputs
I and
represented by the operators
@
correspond
for mutually
the
trivial
effects
1 and O, respectively:
FI = 1 , Second,
to
exclusive
F~ = 0 .
outputs
a and
(6.10) 8 the probabilities
~ 8 -- "either a or 8" to be "yes" must behave addltlvely; t r ( F a , 8 W)
for
i.e.,
tr(FaW) + tr(FsW)
=
for arbitrary states W, which implies F This
can be easily generalized
= F
+ F8 .
to disjoint
(6.11)
unions
of n > 2 outputs
in
the form F
= F al~
Finally,
since
~Aa'
...W~ a n
+ ... + F ~i
.
(6.12)
an
= ~ and e ~ a' = I for all
a, (6.10) and
(6.11)
imply Fa, = 1 - F a = F'a "
i.e.,
F , coincides
with
the
effect
(6.13)
F'e = "not Fa"
introduced
in
§2,
as expected.
For arbitrary outputs a and 8, (6.12) and the relations a = (aAS)~
(=^8')
8 = (aAS)
~(='A
8)
and
av
B
imply a generalization
=
(a ^ 8 ) ~ ( a ^
8') ~ ( a ' A
8)
of (6.11),
Fav
8 -- Fa + F 8 - F= A B "
(6.14)
112
In particular, we get from this F~v Since F A B both
= 0 means
"yes",
(6.15)
B = F~ + F B
(6.15)
the
more
but not vice versa.
F~ A B
-- 0 .
(6.15)
that the outputs e and B are never found to be
has
is somewhat
if
same
physical
general,
background
as
(6.11),
because e A 8 = @ implies
but
F ~ B = 0
We leave it to the reader to derive a generalized
version of (6.15), F
= F ~i v . . . y e n
Since
all
from
(6.10)
+ ... + F el
properties and
characterize F.
if
F
en of
the mapping
(6.12),
these
(Actually,
= 0
for
i # k .
eiAek F : ~ ÷ F
two
listed
requirements
so far follow
are
sufficient
(6.11) and one of the two equations
to
(6.10)
would also be sufficient.)
The
physical
the
interpretation
instance, only
meaning
the
if
of
of
the
effect
both
the effects
e B follows
corresponding
FIA2
F 1 and
F
is
F 2 are
apparatus
triggered
triggered,
immediately
outputs
case
and
of subsystem
F(1A2,)V(I,A2
)
effects, may
in
be
the
apparatus
b
and
may
therefore
be called
however,
represent which
to
mean
well-defined
would
permit
that
the
operators
calculational
rules
(6.13).
If
F~ and FB.
calculate
there
were
the
and
FI, , F l y 2 ,
"F 1 or F2" ,
"or"
for
is
only
effects
corresponding
true
rules
for
also
as
effect
= "F ~ or F~" directly
This
and
and
This notation should not be misunder"and"
F A B -- "F~ and FB" and F v B
e.g.,
"not FI" ,
"not",
to
if
for the particu-
Similarly,
called
"either F 1 or F2", respectively. stood,
§4.)
~ e B. For
on
"F 1 and F2". (We have already used this terminology, lar
from
used
here
operators,
F,
= "not F ",
and uniquely
"not", for
from
according
"and"
and
to
"or",
their successive application - together with (6.13) - would allow one to calculate the
uniquely
existence
already However,
allows
of one
one to
all
operators
additional calculate
F
rule F vB
as we shall prove later on,
~ B from F I and F 2. would from
(In fact
suffice,
since
F A B and
vice
(6.14) versa.)
the effects F I and F 2 do not in
113
general "and"
uniquely nor
determine
" o r " , when
used
rules for (coexistent)
The
preceding
There
exists
B
(6.5)
yields
a mapping
(6.10)
and
generated to
that B
above,
F
represent
e B. Thus
unique
neither
calculational
the following
necessary
condition
for
of two effects F 1 and F2:
tisfying bra
as
effects
effect operators.
discussion
the coexistence
the remaining
(6.9))
contains
F : ~ E B
(6.12),
by
two
onto
a
+
g B,
of the Boolean
elements set
the effects
under F of the particular
F
B
I
of
and
alge2
(cf.
effects,
F 1 and
F 2 as
elements
sa-
such
(6.16)
images
1 and 2 of B,
respectively.
The
Boolean
generated
algebra
by
two
independent
ments.
Condition
the
any more
F I and that
any
as
at
- when
tation
the
least
suitably together
by assuming
le",
as
theorists
also
as
sufficient
completion
The
choice
more
2,
its
apparatus
condition
extended
is motivated
for
tion
has
ever
by
been
abstract
internal
the
F
the
(6.16)
thus
coexistence may
following
proposed.
such
(6.16)
to of
to mean
apparatus, outputs,
b
(at least "in princip-
FI
condition
and
and sufficient
Second,
-
this interpre-
F 2.
(6.16)
The
set
b.
coexistence
First,
no other
has
simple
a
of
as a coexistent
by the apparatus
arguments.
ele-
that such an apparatus
consider of
an
~ e B as
then be interpreted
as a necessary the
of
constructed We
did
interpreted
the elements
to imply
its
measurement
~ B. We go beyond
(6.16)
of
being
one, not refer-
joint
can be
algebra
structure
coexistence
the
existence
- has
add).
of
for
of the set __{FI,F21 , as realized
tion
the
interpretation
(6.16)
but really
usually
of (6.16)
as
definition
all effects
in
here
specific
original
condition
B occuring
and
imagine
can not only be imagined
effects
I
of a measuring
can
and measures
considered
is thus a purely mathematical
F 2. Nevertheless,
one
which
of
(6.16)
-
existence
is
elements
indeed
ring
B
condicondi-
physical
114
background, ably
and its mathematical form can also be simplified consider-
(see below),
(6.16)
so that it is simply applicable
is satisfied,
one may explicitly
as well.
construct
of an apparatus
b for the joint measurement
all
effects
Last
sion
effects,
criterion
of
F
e B.
(6.16)
but not least,
reproduces
conventional
specific applications
the
quantum
of FI,
when applied
well-established
mechanics.
of (6.16), however,
if
a quantum mechani-
cal model other
Third,
Before
F 2 and
to decl-
commutatlvlty
discussing
such
we shall first transform i t
into an equivalent but much simpler conditon.
In this connection the four particular elements i A 2, of
B,
play
listed a
unique kind:
above
decisive way
I A 2',
under
rSle.
1'1% 2,
(6.6)
Every
as a disjoint
and
excluding
element
e e B
each
can
be
union of n 4 4 elements
n = 0 yields ~ = @; n = 1 yields
selves;
i' I% 2' other
represented
in a
of this particular
the four elements
for n = 2 we get the two elements
palrwlse,
(6.6) them-
(6.7) and the four elements
(6.5) - the latter, because (IA
2) ~ , * ( I A
(IA
2) , ~ ( 1 ' /X 2) :
2')
:
1 A(2
V 2')
= 1A
I
:
1 ,
(6.17) (i V I')
A2
:
I A2
: 2 ,
etc.; for n = 4 we get (I ^ 2) ~ ( I A 2 ' ) :
(I A
(2 V 2 ' ) )
%>(I'A V
(I'A
2) ~ , ( I ' A (2 V 2 ' ) )
2')
: 1Vl'
: I
(6.18)
,
so that, finally, (I A 2) ~ (I A 2') ~ ( I ' A etc.; the
i.e., last
n = 3 yields
equation,
implies B = ~'.)
we
2) = ( l ' a
the remaining
have
used
(6.18)
2')' = 1 V 2
four elements and
the
fact
,
(6.8) of B.
(In
that e ~ ' 8
= I
115
Consider
two effects F 1 and F 2 satisfying
the coexistence
condition
(6.16). Then there exist four effects, FI2 = FI A 2
'
FI2' = F I A 2 '
'
1 (6.19)
FI, 2 = F I , A 2
,
FI'2' = F I ' A 2 '
the images under F of the four elements (6.18),
together
with
(6.10)
and
J
'
(6.6) of B. Eqs.
the additivity
(6.17) and
property
(6.12)
of
the mapping F, imply
FI2 + FI2, = F 1 ,
(6.20)
FI2 + F1, 2 = F 2 ,
and + FI2, + FI, 2 + FI,2, = 1 .
FI2 More generally,
since an arbitrary element = ~ B is a disjoint union
of n ~ 4 elements ~i from ding
effect
mapping
F
(6.21)
is
the
F : B ÷ B
(6.19) are known.
is
(6.6),
(6.12)
of
n
sum
thus
the
completely
Actually
implies
effects specified
that the correspon-
F~i if
from
(6.19).
The
the four effects
it suffices to know three of them,
e.g.,
FI2, FI2 , and FI, 2. They satisfy, because of (6.21) and FI,2, ~ 0,
(6.22)
FI2 + FI2, + F1, 2 < 1 , and the missing fourth effect FI,2, is obtained from (6.21):
(6.23)
FI,2, = i - (FI2 + FI2 , + FI,2) . (Eq.
(6.23)
immediately
implies
FI,2, < I and - with
(6.22) - also
FI,2, ) 0; thus it really defines an effect.)
We
have
according and
thus to
shown
that,
condition
FI, 2 satisfying
if
two
(6.16),
(6.22),
terms of these three effects
effects
there
such
exist
F 1 and three
that F 1 and
F 2 are effects
F 2 may
coexistent FI2 , FI2 ,
be written
in
in the form (6.20). Assuming that, con-
116
versely, with
three
prove
For
two
effects
that
purpose,
(6.21)
FI,2, ,
in
elements
is
shown
(6.6)
These
the
way
operators
as
Eqs.
(6.10)
trivially
effects
they
FI
satisfied
have
and
all
belong
The
F2,
that
of
the
to L(H). and
to
respectively.
As
F
validity whereas
for
then - as ~ ~ B (6.19). arbitrary obtai-
of n ~ 4 diffe-
of
the
is
effects 1 and
indeed,
the
first
the second
(6.12)
elements
Thus,
of the type
F : ~ ÷ F
sums
to
particular
elements
F
so
are > 0, and also ~ 1 by
condition of
FI2
effects
define
the mapping
(6.21),
the
four
particular
The
construction
F also maps
effects
remaining
properties.
additivity
explicit
four
now
(6.23),
F exists,
operators
(6.18)
four
by
a mapping
taken
to show
required
from
(6.20),
be
FI,2,
E L(H)
the
(6.20)
we shall
: ~ ~ B ÷ F
of
the
now
(6.19),
the
and
of
F
form
(6.16).
the
images
If such
under
sums
the
satisfied.
(6.17)
lowing
from
of
F
remains
follows
consequence
a mapping
can
has
therefore
as
F
effect
interpret
(6.19),
(6.22),
condition
a fourth
we
(6.16).
images
it only
this
(6.21);
We
with
representations
in
Then
of B under
-
and
define
satisfied.
in the
FI, 2 satisfying
the coexistence
first
represented
E B,
by
we
in condition
above be
rent
FI2 , FI2 , and
accordance
considered
may
F 1 and F 2 can be represented
that F 1 and F 2 satisfy
this
ned
effects
an
of
one is
immediate
F . Finally,
2 of B into
condition
(6.16)
the is
for F 1 and F 2.
thus
proved
simpler
that
coexistence
Two
effects
if
they
F I and
can
be
condition
(6.16)
is
equivalent
to the
fol-
criterion:
F 2 are
coexistent
represented
in
the
if and only form
(cf.
(6.20)) F 1 = F12 + FI2, in
terms
tisfying
of (cf.
three
,
effects
F 2 = FI2 + F1, 2 FI2 , FI2, and F1, 2 sa-
(6.22)) FI2 + FI2 , + FI, 2 ~ 1 .
(6.24)
117
The
physical
F1,2,
(the
(6.19), above, F2",
interpretation
latter
and
is
being
most
according
of
the
defined
easily
four
by
effects
(6.23))
expressed i n
FI2 , FI2,, F1, 2 and
follows
the
immediately
terminology
from
introduced
to which they may be called "F 1 and F 2 , "F I and not
"F 2 and not FI", and "not F 1 and not F2" = "neither F I nor F2" ,
respectively. single that
If,
therefore,
microsystems single
tr(Fl2W),
by
a
the
and
suitable
microsystems
wheras
FI
in
sense,
measured
W
trigger
for
the
then
operators
the
both
triggering
and
but
In
the correlations
between the results of joint measurements
effects F 1 and F 2. Generalizing
on
probability
FI
F1
F12
together
tr(Fl2,W) , etc.
coexistent
this
are
apparatus,
state
probability
F2
not
to FI,2,
F2
is
F2
is
describe
of the two
a terminology already used
in §4, we therefore call them correlation effects.
The
three
uniquely
correlation
determine
effects
F12 , FI2 , and
all effects
F2
besides
are
given
in
as
F , = s B, and thus,
the two effects F I and F 2 (cf° and
FI,2,
(6.20)).
advance,
it
among
If, as usual,
therefore
also
shown
above,
them,
also
the effects F I
suffices
to
know
them only a single one of these three correlation effects, as
the two others may
then be calculated
from
(6.20).
Since Eqs.
(6.20)
and (6.21) also imply
F 1' = FI, 2 + F1 ,2'
the knowledge
of FI, F 2 and FI,2,
effects
F
Before
drawing
general
criterion
application
of
performed and
(6.25)
,
is also sufficient to calculate all
.
coexistence
denoting
F 2' = FI2, + F1 ,2'
'
two
(6.24)
effect
the selective by
F 2 = ~*G,
f,
the
by
the
aparatuses
we
shall
example
two
effects to
(6.1).
(6.1),
operation,
measured In
first
illustrate the
the
successive
f and g. With ~ and $ = ~ + ~'
and non-selective
according
fects (6.19) are
conclusions,
this
together case,
respectively, are
as
F 1 = F = ~*i
the correlation
ef-
118
FI2 = ~*G
To
show
the
,
this,
consider
apparatus
tr(FW)
FI2 , = ~*G'
f
= tr(~W).
,
FI, 2 = ~'*G
microsystems
-
i.e.,
Those
FI,2,
in a state
the
systems
,
effect
which
have
W.
F1
= ~'*G'
They
-
first
with
triggered
. (6.26)
trigger
probability
f go into
the new
^
state
W = ~W/tr(~W),
i.e.,
the
the
effect
successive
of the effect lities, (We
fled,
F2
-
afterwards
probability
of
both
(6.26)
= tr(G.~W)
presented follow
this
f and
argument
in
Eqs.
g
probability
for
the
-
for
occurrence
of these two probabi-
This §2.)
(6.20)
apparatus
The
g - i.e.,
= tr(~*G.W).
similarly.
the
tr(GW).
"F I and^ F2" = FI2 - is the product
already
in
triggering
with
triggering
tr(~W).tr(GW)
have
tions
thus
implies The
and
FI2 = $*G. ~
remaining
(6.21)
are
equasaris-
since FI2 + FI2 , = ~*(G + G') = ~*I = F = F 1 , FI2 + FI, 2 = (~* + ~'*)G = ~*G = F 2 ,
and FI2 + FI2 , + FI, 2 + FI,2,
= ~*(G + G') + ~'*(G + G') =
(Alternatively, last
three
as remarked
We
shall
show
and
use
Eqs.
effects
in
+
(6.20)
~''i
and
(6.26)
=
F +
F'
(6.21)
from
FI,
=
i
.
to calculate F 2 and
the
FI2 = ~*G,
above.)
derive
from
(6.24)
some
general
results
on
pairs
of
effects.
i__t. Two effects
To
could
correlation
now
coexistent
we
~*i
this,
F 1 and F 2 with F 1 ~ F 2 are coexistent.
take
FI2 , = O.
FI, 2 = F 2 - F 1
F 2 • I), possibility
and of
(6.22)
(the is
choosing
Then,
latter valid
by
(6.20),
being
since
~ 0
we
must
since
set
FI2 = F 1
F 2 ~ FI,
FI2 + FI2 , + FI, 2 = F 2 • I.
FI2 , -- "F 1 and not F2" = 0
means
that,
and The on
a
119
suitable rence
apparatus
of
F 1 is
that apparatus
2__t. Two
always
F 1 and
F 1 < F~
joint measurement
accompanied
FI2 = 0 and
exclude
equivalent
described
To show this,
by
other
(i.e.,
FI2 , = FI, Clearly,
never
each
and
FI2 , = FIF ½ ,
easily
other,
the
checked. same
also for arbitrary
joint
measurement
zed
as
successive
First,
is
effects
The
above:
is
that
thus
the
symme-
F1, 2 = F2;
then,
means
together)
on
indeed,
that an
F1
and
apparatus
= 0 are coexistent.
FI, 2 = F;F 2 ,
are > 0 and < 1 since,
< ilfll~ = (f,lf)
is
a
on
of F 2.
(Note
and
FI2 = 0
occur
[FI,F2]
apply
of
for As
all the
true F
for
= FIF~
.
(6.27)
e.g.,
f E H. four
The validity
operators
arbitrary
of
(6.27)
sums
of
(6.20)
commute
them,
and
and with thus
c B.
F 1 and
measurement, an effect
FI,2,
112 112 fl;2 = IIF1 F 2
(f,FiF2f)
(6.21)
coexistent.
i.e.,
set
operators
~ 0
of F2;
the occurrence
to F I + F 2 < I,
(6.20),
F 1 and F 2 with
FI2 = FIF 2 ,
is
occurrence
the occur-
by this choice.
3__t. Two effects
These
of F 1 and F2,
to F 1 and F2.)
thus,
each
the
F I < F~ are
FI2 + FI2 , + FI, 2 = F 1 + F 2 < I. F2
by
of F 1 implies
F 2 with
is
tric with respect
Set
the
the occurrence
effects
relation
for
F 2 described as
in
apparatus
the
by
(6.27)
example
fl performing
may
(6.1)
be
reali-
considered
the complementa-
ry operations
: W
and
thus
+
measuring
_1/2 _1/2 ~I W~2 '
the
effect
~'
: W
~'1 = FI;
+
after
,i/2 ,1/2 F1 W FI ,
this,
apply
(6.28)
an appara-
120
tus
f2
commute,
measuring
F 2.
Then,
by
(6.1)
and
the combined apparatus measures
the
fact
that
F I and
F2
together F I and
1/2 I/2 F~1/2F2F~1/2 ~*F 2 = ~*F 2 + ~'*F 2 = F 1 F2F 1 +
= (F I + FI)F 2 = F 2 ,
as desired,
while
(6.26) - with G = F 2 and ~,
#' from
(6.28) - imme-
diately leads to (6.27).
The converse of statement 3 is not true, however:
4__t. The
operators
F I and
F2
describing
coexistent
effects
need
not
E
with
commute.
As
an
example,
consider
two
projection
,
F 2 =~E+~-
operators
E
and
[E,E] ~ 0, and set I
F I =~E (Since
.
(f,F2f) = (f,Ef)/2 + (f,Ef)/2 < llfll2 for all
an effect,
f s H, F 2 is also
i.e., ~ 1.) Then we have 1 [F I ,F 2] = ~ [E,E] # 0
but as F 1 ~ F2, F 1 and F 2 are coexistent
Commutativity
is necessary
for
of the two effects considered
5__t. A decision and
only
coexistent ting;
effect if
completion
F12 = EIF 2 ,
coexistence,
however,
if at
least
one
is a decision effect:
E I and an arbitrary
[EI,F2] = 0.
in particular,
by !.
In
B of
this
{EI,F2}
effect
case,
all
F 2 are coexistent operators
are unique
F
and mutually
in
if the
commu'
the correlation effects are
FI2, = EIF i ,
' 2 , F1, 2 = EIF
' 2' • FI, 2 , = EIF
(6.29)
121
To show this,
we first prove a mathematical
Lemma:
be
Let
E
satisfying
Proof:
a
an
llF1--~fll2 = (f,Ff)
i.e.,
A
Since
arbitrary
< (f,Ef)
E(E'g)
= 0
vector
= 0,
then
of
E1
choice
the
and
of
be
and
for
statement
all
F2
the
are
~
coexistent.
FI2 < El, so that,
g E H,
implies
already
coexistent,
correlation
f e H
i.e.,
FE' = F(I - E) = O, which
part
to
operator
F
an
arbitrary
operator
0 < F < E; then EF = FE = F.
Consider
Ff = 0.
projection
statement:
with
also
we
this
F(E'g)
= O,
get
from
FE = F = F* = (FE)* = EF.
follows
from
effects
(cf.
(6.27)).
by
(6.20),
~:
If
[EI,F2]
represents Now
a
assume
E 1 = FI2 + FI2,,
possible E l and F 2 and
(cf.
(6.25))
.
leads to
= FI, 2 ,
with EIE; = EIE 1 = O, we get
(6.30) imply
tions
mutual
As
and
(6.31)
all
follow
from
other
effects
commutativity
a particular
criterion
show
EIF 2 = FI2 , the
then
(6.29),
thus
(6.30)
EIFI, 2 = FI,2E 1 = 0 .
and
= O,
by the lemma,
EIFI, 2 = FI,mE;
Eqs.
get
thus
(6.29)
E i = FI, 2 + FI,2,
from which,
we
and
EIFI2 = FI2E 1 = FI2
Similarly,
Then
F1/2f = O,
and
Then,
Ef = 0.
first
of "ordinary"
of ~,
F
El
commutes
of Eqs.
(6.20)
follows
case
that
and
E B
are
(6.31)
with
(6.29).
(6.21). also
F 2 = FI2 + F1,2,
The other
With
uniquely
the
three four
determined.
equa-
effects Their
as in ~.
we
obtain
the well-known
quantum mechanics:
commensurability
122
6.
Two
decision
[EI,E2] sion
= 0.
effects In
effects
determined
B
is
a
phism,
In
case,
by E 1 and
are
consists
With
the
EaVE
Boolean
algebra,
F
and
only
commuting
= E , which A,
are
V
if
deci-
uniquely
and
' defined
by
~ = E a + EB - EaE B ,
and
if
of mutually
operations
operators
= EaE B ,
coexistent
the
mapping
E'a = 1 - Ea
F : B + B
'
is
(6.32)
a homomor-
i.e.,
first
= EaA
statement
the particular
described
EB
and
case
by projection
,
the
They
are
not
only
i.e.,
EI2EI2 , = 0, onto
considered,
of
the
B ,
all
Ea, = E'a "
F
e B
correlation
(6.33)
follow
effects
from
(6.29)
5. are
operators,
,
mutually etc.,
mutually
E v 8 = EaVE
uniqueness
FI2 = EIE 2 = El2
project
B
E2
operators),
E 2.
projection
EaAB
The
and
(projection
for commuting
EaAEB
this
E1
FI2 , = EIE ~ = El2 , , etc.
commuting
which,
as
orthogonal
but
is
even
well
subspaces
(6.34)
mutually
orthogonal;
known,
means
that
they
of
Furthermore,
Eq.
H.
(6.21),
El2 + El2 , + El, 2 + EI,2,
means
that
H
Since
all
operators
operators are
also
sum of
is
from
the
E
~ B
(6.34),
they
projection
the
direct
ranges
sum of are
sums
likewise
operators
of
the
E
-
four of
= 1 ,
subspaces
n < 4
commute
each
the n projection
E
different among
projects
operators
EI2H to EI,2,H.
from
projection
themselves, onto
the
(6.34)
and
direct
which
add
up to E .
Consider sented
now as
two
such
disjoint
operators,
unions
of
Ea
and
elements
E B. of
When B
from
~ and
~ are
(6.6),
E
repreand
EB
123
become can
analogous
easily
be
sums of the corresponding
seen,
a ~ B is
the
operators
disjoint
union
from
of
(6.34).
those
As
elements
from (6.6) which occur in both a and B, whereas a v B is the disjoint union of those elements representations
of
a
from (6.6) which occur in at least one of the
and
B.
On
the
other
etc., E E B is the sum of those
operators
both
EB,
sums
representing
E
and
and
hand,
from thus
since
(6.34)
in E , E B or both of them,
which
coincides
whereas E a + E B - E E B is the sum of those operators occur
E12EI2 , = 0, occur
with
in
E A B,
from (6.34) which
thus being equal to E v B. (EaE Bhas
to be subtracted from Ea + E~ in order to avoid doublecounting of the operators
(6.34)
which
(6.14).) With A , tors,
we have
V
occur
and
in
both
E
and
E B.
Compare
also
Eq.
' as defined by (6.32) for projection opera-
thus proved
the first two equations
in (6.33), whereas
the last one already follows from (6.13). The preceding argument also yields
the known geometric
defined
by
(6.32): E
E H ~ EBH E
of
AE B = E E B
the
two
+ E B - E E~
V E8 = E
by E H and EBH.
interpretation of the operations projects
subspaces
projects
onto
As is also well known,
onto
E H
the
and
~
and
v
intersection
EBH,
whereas
the subspace E H + EBH E'e = 1 - Ee projects
spanned onto the
orthogonal complement of the subspace E H.
The Boolean algebra and
'. By
with
(6.33),
respect
applied
to
B is closed with respect then,
the
to mutually
to the operations
the set of projection
analogous commuting
operations
projection
operators
defined
by
operators,
A ,v
B is closed (6.32).
these
When
operations
(6.32)
are well known - and easily checked - to satisfy the calcula-
tional
rules
(6.3)
and
(6.4) of a Boolean algebra,
with the rBles of
and I taken by the operators 0 and I, respectively;
(E
A EB)' = i - EaE B = (i - E ) + (I - E B) - (I - E )(I - E B) = E'a + E~ - E'E'~B = E'a V E~
By
(6.10)
algebra
e.g.,
and
(6.33),
structure;
the
i.e.,
it
mapping is
a
F : B + B
.
preserves
homomorphism.
As
the
a % B
Boolean does
not
124
necessarily
imply
E
# E~,
(i.e.,
an isomorphism).
It
is
also
(6.32)
for
"or"
and
well
known
coexistent "not",
the mapping
that
the
decision
F is not
operations effects
respectively.
This
A,
may now
be
and the physical m e a n i n g of the effects
7__7_.In
general,
the
two
set
of
According nor
F2
consider
coexistent effects
pletions
F , ~ ~ B
effects
{FI,F2}
may
F1
are
and
have
and
F2;
several
' defined
interpreted
E
not
one-to-one
"and",
immediately
^~'
E
vB
uniquely
i.e.,
as
by
a
and E ,.
determined
given
different
from
by
coexistent
coexistent
com-
B.
to is
effects
V
follows
(6.33)
the
in general
~,
a
such
decision
non-uniqueness effect.
is
As
a
possible
very
only
if
instructive
neither
example
F1 [I],
the effects 1 FI = ~ E ,
with a "nontrivial"
projection
I E' F2 = ~ E +
operator
E (i.e, E # 0 or I).
i)
!
Since F 2 -- FI, F I and F 2 are coexistent,
FI2 = 0 ,
(6.35)
and we may take
and thus FI2 , = F 1 = ~ E ,
(6.36) FI, 2 = F2 = ~i E + E'
according
to ~.
On the
other hand,
F 1 '2' = 0 ,
,
since
F I < F 2 as well,
we may also
set
F12 , = 0 ,
and thus FI2 = F I = ~ E , (6.37)
FI, 2 = F 2 - F 1 = E'
as in i- Finally,
since
[FI,F2]
,
FI,2,
1 = ~ E ,
= 0, a third possibility
is to take
125
I
I
F12 : FIF 2 = ~ E ,
FI, 2 = F~F 2 = F 22 = ~1 E + E' as
in 3. For
(6.16)
the example
may
be
F : ~ e B + F
e L(H).
-
not
only
for
~ = 1A
Indeed,
in
yielding 2),
but
as can easily
,
with
These
three
mappings
effects
ranges
be proved,
the coexistence
least)
three
different their
therefore, (at
for
the choice
(6.37)and
(6.38)
from
suitable
B = FB c L(H)
are
(6.36)
condition
different
differ
F
i I B : {0, ~ E, ~ E + E', wheras
(6.38)
, 2, = F2FI = ~1 E , = FIF
FI,2,
(6.35),
satisfied
1
FI2 , : FIF2 = ~ E ,
each
other
~ ~ B
(e.g.,
also
yields
mappings
different.
the set
I} ,
lead to
B = {0, ~1 E, E, E' , ~I E + E',
i1
and 1 I 3 1 I 3 B = {0, ~ E, 5 E, ~ E, ~ E + E', ~ E + E', Z E + E',
I}
,
respectively.
As
shown
the of
before,
correlation them
if,
also
given.
this
simply
suits
of
depend
not
lar
the mapping
as
means
joint only
used
uring
apparatus.)
lence
classes
should
and
it
of F
should In view
rather
be too
even
here,
statistical of
effects
the be
FI
suffices
them
realizable
of the fact
by F I and F 2,
of
the
on
between
single
the
re-
microsystems
but also on the particu(Note
by means effects
that,
(6.16), of
according every
a suitable
F represent
apparatuses
correlations
surprising;
on
condition
effect
one
determined
F2
that
specify
F I and F 2 are
together.
coexistence
to
correlations
and
by specifying
the effects
themselves,
than particular
dependence not
the
for measuring of
determined
F is not uniquely
measurements on these
choice
fects
that
interpretation
apparatus
(6.19),
therefore,
sible
ble
is completely
in the case considered
If,
apparatus
to our
effects
F
between
the contrary,
posmeas-
equiva-
f, such possicoexistent
one might
ef-
rather
126
be surprised these
that in particular
correlations
turn
out
cases,
to
be
as specified above in 5 and 6,
independent
of
the
choice
of
the
measuring apparatus.
For
our example
ferent
mappings
choice
(6.36)
possibility and
(6.35), F
listed
of
measuring
ratus.
Then,
F12 = 0,
by
as
apparatus
for
effect
E 2 = E'.
each
other
to
the
together
To
joint
These
FI
the
effect
means
"F 1 or E2"
"either F 1 or E2",
F 1 + E 2 = E/2 + E' = F 2. FI
implies
accordance
the
by
thus
of
-
the correlaions
between
consider
and
the
and
an
decision
they exclude
apparatus
This a p p a r a t u s
since
also meas-
"F 1 and E2" = 0
-
actually
Eq.
(6.15))
definition,
the
occurrence
of
"F I or E2" = F 2 at
this
apparatus,
in
by
F12 , = 0 in
(6.27), of
consequences
i.e.,
(cf.
the choice
remarkable
F1
FI, and
other,
(6.37),
chosen
as
with
application
each
by ~,
coinciding
Moreover,
of
effects
of F 1 at this appa-
F 1 = E/2
(6.29).
of
case
choice
The
mentioned
fl measuring
exclude
the
which
uring F I and F2, respectively,
Some
F2
occurrence
successive
repeatedly
apparatus
the
as a particular the
complementary
two
coexistent,
of
easily.
the
and
are
quite
and
measurement
irrespective
effect
visualized
the three dif-
obvious
realize
"F I and E2" = FIE 2 = EE'/2 = 0, ures
realizing
of F 2 as the non-occurrence
(6.36). the
be
effect
definition,
in
can
an arbitrary
the occurrence
apparatuses
above
corresponds
F 2 = F 1'". take
define
suitable
(6.37).
the choice suitable
(Compare
!.)
with
Finally,
(6.38) may be realized
apparatuses
fl
and
by
f2 meas-
as explained in 3.
of
the possible
coexistent
effects
apparatus
dependence
of
may also be illustrated
by
the example (6.35).
According
to ~,
on a suitable
two effects measuring
another
apparatus
mentary
effects
apparatus does
not
is
F 1 and F 2 with F 1 < F~ exclude
apparatus.
used.
As
an
F I and F 2 given
described exclude,
by but
(6.37): rather
This
need
example,
by (6.35), In
this
implies
not
be so,
consider
the
as measured
case,
the
the occurrence
each other however, two
if
comple-
together by an
occurrence of F 2.
of
FI
(In view
127
of
such
possibilities,
the
effect
F'
the
obvious
it might
by "not F".
and most
appear
However,
"natural"
two
tus !.
effects
the
this,
this
take
described
again by
not
the
be
(6.36),
suggestively
of measuring
F'
together
then on a suitable
the occurrence
true
effects
quite
to denote
(6.35) by the choice (6.36).)
of F 1 implies
need
misleading
expresses
F 1 and F 2 satisfy F 1 ~ F2,
occurrence
Again
this
possibility
with F, as realized in the example
If
a little
for
other
according
apparatuses.
F I and F 2 from
the occurrence
of F2,
(6.35):
of F I excludes
appara-
To
to
exemplify
On an apparatus the occurrence
of
F2, rather than implying it.
Two
identical
to i, this (as
we
effects,
may
case.
choose
On
an
FI2 , = 0),
occur the
output
channels. uring one
an
an
implies
described
versa
by
can
this
be
f measuring
coexistent.
F12 = F and
(as FI, 2 = 0);
apparatus
apparatus
always
choice, i.e.,
realized
F
into
same
effect
apparatus:
corresponding correlation
Consider to
again
(6.37)
effects
but
F by reading
rather
for
FI2
and
effects
their
joint
FI,2,
two
than always
occuring
F 1 implies
by feeding
however, output and
measurement. coincide
output of meas-
channels an
of
apparatus
Then
with
together,
F2
F 2 always
different
(6.35)
both
FI, 2 = 0 in
simply
two different
the
According
F 1 and
There are also less trivial possibilities,
the
F = E/2,
vice
Such
of
are
FI2 , = 0, which
apparatus
and
together.
F 1 = F 2 = F,
the
the
two
effect
they actually
ex-
clude each other.
Such
things
effects
FI
E I < F½,
can and
F 2 < El,
so
that
happen,
F2
E I and
to
not
is
a
F 2 always
therefore
the
however,
decision
E I if F 2 ~ El,
and E 1 implies
lemma
in
~
yields
i.e.,
F2
implies
EIF 2 = F 2 E1.
In
the
each
say
each other.
F 2 if E l < F 2. that,
second
one
by
case,
of
F 1 = E l.
EIF 2
In particular,
exclude
so
least
other: E l < F½
in ~ implies
FI2 = EIF 2 = 0, by (6.29).
sion effects E and E' always
at
effect,
exclude lemma
if
is
the
two
Then,
if
equivalent
(i - EI)F 2 = F2, complementary
Moreover,
In the
deci-
F 2 implies
first
case,
the
(6.29),
FI, 2 = E~F 2 = 0;
E l ~ F2
yields
F½ ~ E~,
128
so
that,
again
by
!
the
F12 , = EIF 2 = 0;
i.e.,
forms
when
together,
measurements of
all
of
these
ment.
Indeed,
spond
to
one
E 1 implies applied
and
the
measurements if
' 2' = F 2' EIF
lemma,
two
measurements
F 2. Finally,
to
must
of
a
same
outputs E,
FI2 , = FI, 2 = EE' = 0 which,
and
single
if some
effect
identical
some
i.e.,
F I = F 2 = E,
two
then
then
per-
or
more
the
results
single
experi-
apparatus
explained,
(6.29),
apparatus
every
2
already
E,
in
I and
as
of
by
microsystem,
decision be
thus,
both
corre-
(6.29)
yields
implies
the desired
conclusion.
The
last-mentioned
I.e.,
if F
apparatus systems
is
not
which in
such
in
that
to
the
a way,
ble
W.
measurements always
preceding
respect,
We of
give
identical
E
turn
like
"properties"
measurements
According
the
vial)
to be of
it
is
that
at
least
effect. following
projection
This
replaced
ger estimate
be
the
F
on single results show
in
obtained
micro-
of
these
replace
correlation
effects
the
first
and
"no"
so in
are non-zero
for suita-
succinctly:
Different
more same
by
an
this,
microsystem
is a decision
itself,
need
effect.
measuring
dependent,
microsystem
not
In this
decision
ef-
and thus appear more than
the
uniqueness
of
results
of
F.
one
E
[FI,F2] here,
for
of
Take
operator
(6.22)
on
unless
sufficient
example.
Since
result
F
condition
0 < ~ < ~ < I. may
this
exists
FI2 , = FI, 2 = F - F 2 # 0,
"yes"
less apparatus
!,
the
effects.
there
the To
by tr(Fl2,W),
results
the
other.
yields
effect
of other effects
to
F : ~ ÷ F
by
out
the
each
obtaining
results
decision
then
sometimes
a n d choose
now
as given
same
(F 2 ~ F),
least
argument,
for
for
two F measurements
from
express
the
fects
at
this
may
therefore,
decision
that
measurement,
states
effect
different
(6.27);
characteristic
together
probabilities
the second
is
decision
are
the
according
a
performs
two measurements E by F
property
is
the
the not
F 1 = ~E and
real
two
necessary, and
(6.20)
FI
F 2 are
~
mapping
and
F2
is
however,
as
shown
F 2 = BE'
numbers
= 0, F I and using
effects
the
with and
~
coexistent.
and FI2 > 0,
a
a
(nontri-
satisfying Condition
by the
stron-
129
(6.39)
FI2 + FI2, + FI, 2 < F I + F 2 = ~E + BE' < B(E + E') : ~I
The
effects
G I = FI/B = ~E/B
and by (6.20)
and
G 2 = F2/B = E'
they may be written
G 1 = GI2 + GI2, with correlation
are
also
coexistent,
in the form
,
G 2 = GI2 + GI, 2
effects GI2 = FI2/B , etc., which indeed satisfy
G12 + GI2 , + GI, 2 < 1 , according
to
(6.39).
correlation
effect
correlation
effect
But G12
because is
G 2 = E'
unique
F12 = BG12
is
(cf.
is
also
a
decision
6).
Therefore
unique,
which
effect, the
the
original
implies
unique-
ness of F.
8. Consider,
as in §4,
a composite
acting
subsystems
effects
(cf. (4.10)),
are
coexistent,
correlation
I
and
system
II.
Then
consisting
arbitrary
of two noninter-
pairs
~i = FI @ III '
~2 = II ~ FII '
according
and
to
3,
a
of
subsystem
(6.40)
possible
choice
is
the
FII only
(and
FII
thus
possible
F : B ÷ B C L(H). (6.41)
decision
therefore
most
subsystem
effects
Fl
(6.41)
or F2 ) is a decision
choice,
Moreover,
is in a certain
is a
the
effect ~12 = "El and ~2" is, by (6.27),
~12 = ~I~2 = FI ~ FI! "
If F I or
for
"natural" ~I
and
leading
as discussed
sense
effect,
thus
it
possibility ~2"
(See
there occurs as Eq. (4.9) - in §4.)
to
in detail
the "natural"
since
effect, a
then
unique
choice
one also if neither
F I nor
to
the
simplest
for the joint measurement the
mapping
the
corresponds
in §4,
(6.41)
derivation
of
(6.41)
and
of the - which
130
Nevertheless,
there
are
also
pairs
which (6.41) does not represent
of
subsystem
effects
(6.40)
for
the only possible choice. As an examp-
le, set
I ~2 = II @ 2 Ell
~I = ~1 E1 @ Ell ' with two nontrivial
projection
operators
E I and Eli . In this case, we
may take F = ~(E I @ Ell ) -12 with
an
arbitary
real
~12' = ~i - ~12
and
~
between
0
and
~i'2 = ~2 - ~12;
1/2.
Eqs.
these
two
(6.20)
then yield
operators
really
belong to L(H) since, e.g., i ! ) ~I ~ ~I - f12 = f12' = 2(El ~ III) - a(El ~ Eli) i I ' ) ~ 0 ~((m I @ iii) - (E I ~ Ell)) = ~(E I ~ Eli Moreover,
since both ~I and ~2 are < !/2, we have
~12 + ~12' + ~i'2 = ~i + ~2 - ~12 < ~1 + ~2 ~ ! , so that (6.22) is also satisfied.
Besides
illustrating
between
subsystem
may
exist
pair
of
Except
coexistent
"natural" how
effects,
infinitely
for
the
choice
apparatuses
the possible the
many
(6.41)
above
different
effects,
particular
non-uniqueness
as case
example mappings
parametrized ~ = I/4
of
also
here
- which
all
these
mappings
F
correlations
shows
F : B + B by
that for
the
unknown,
would
look
there
a given
number
corresponds
for ~12 - it is completely
realizing
the
to
~. the
however, in prac-
tice.
We
conclude
effects
our
discussion
of
the
particular
F I and F 2 with the construction
case
of two coexistent
of a quantum mechanical model
131
for
their
joint measurement.
bed
as
quantum
mechanical
system
and
assumed
form
state
a Wa,
is
binary
scattering
tor
on
S
--
output have
tors
rather
"read"
microsystem. performed
now
a
be
This
at
Ela and
than the
with
the only
thus
as described
E 2a on
the model
with
H a.
state the
space
by a unitary
a single
is
its of
is descriand
scattering to
yes-no
yes-no
pair
"reading"
is
a
opera-
have
interaction
two
initial
considered
Their
by a coexistent the
Ha
assumed
one.
after
consists
Actually
apparatus
microsystem
apparatus
apparatus
"reading"
together,
in §5,
characterized
H = H @ H . But
channels, to
to
system,
--
As
two
outputs with
the
measurements
of effect performed
operaon
the
composite system, and is therefore described by the two effects a a I ~ E 1 and I @ E 2 on H. As is obvious, e.g., from the condition a a (6.24), the latter are indeed coexistent if E l and E 2 are. (The a
converse sion
is equally
effects.
assume
this
obvious
Although,
as
- e.g.,
from 6 -
notation already a a the moment E I and E 2 may
later
on,
at
apparatus
of
this
a
if E l and E 2 are
our
suggests, still
deci-
we
will
be arbitrary
effects.)
A
model
F 2 of
the
microsystem,
kind
defined
measures
together
implicitly
but
two
effects
uniquely
by
F 1 and
the
equa-
tions tr(FiW)
with
W ~ K(H)
right
hand
after
the
W.)
arbitrary.
side
of
(6.20),
the
(Compare
(6.42)
interaction
As expected,
represent
= tr(l @ E~)S(W @ Wa)S*) ,
of
F 1 and
coexistent
the apparatus F 2 are
remember
for output
with microsystems
coexistent.
"output"
and
In o r d e r E al
effects
and
(6.42)
that
i to be "yes" in the state
to prove a E2,
the
in
this,
we
analogy
to
in the form
four
lations
(5.1),
is the probability
a a a E 1 = El2 + El2, with
Eq.
i = i or 2 ,
effects
between
a
,
a
a
E 2 = El2 + El, 2 ,
(6.43)
E? • (i = I or i' , j = 2 or 2') describing the correlj the two "output" effects E al and E 2a and satisfying,
132
in analogy to (6.21),
a + El2 a , + El, a 2 + E ~ '2' = la . El2
Corresponding
effects
F.. lj
of
the
microsystem
can
(6.44)
then
be
defined,
analogous to (6.42), by
tr(FijW) = tr((l @ E~j)S(W @ Wa)S*) ,
These
i = 1 or i', j = 2 or 2'.
(6.45)
effects
F.. ij have to be interpreted as correlation effects for F I and F 2 (cf. (6.19)) since, e.g., with i = 1 and j = 2', the right hand
side
apparatus -
by
of
(6.45)
effect
definition
for
the
occurrence
a , = ,,Ka Ka,, after El2 -i and not -2
the
interaction,
of
is
the
occurrence
of the effect
FI, F 2 and
the
(6.43)
the
probability
effects
F I and
F2 -
is
"F I and not F2". Eqs.
correlation
effects
Fij
and (6.44) and the definitions
are
equivalent
of
the
which to
the
(6.20) and (6.21) for
easily
(6.42) and
derived
(6.45).
from Eqs.
(For instan-
ce, (6.42), (6.43) and (6.45) lead to a
tr((F12 + FI2,)W) = tr((l @ [El2 + EI2,])S*) = tr(FiW)
for all W, which implies F 1 = FI2 + FI2,.)
Now consider, stent
conversely,
effects,
an arbitrarily given pair {FI,F2}
and a representation
(6.20)
given - but, in cases of nonuniqueness, tion effects and
the
apparatus
Fij.
given of
the
sion,
this amounts
state
W a g K(Ha),
E?lj Eqs.
e L(Ha) (6.45)
type
effects
considered.
to proving a unitary
(i = I or I' are
of F 1 and F 2 in terms of
deliberately chosen - correla-
We will prove that the joint measurement
correlation
the existence operator
with
of FI, F 2
F.. can be "realized" by a model 13 According to the previous discus-
_S on
j = 2 or 2')
satisfied
of coexi-
the
of a Hilbert H 8 Ha,
and
space Ha, a four
satisfying
(6.44),
given
correlation
effects
such
that
effects
F.. e L(H) and arbitrary states W ~ K(H). With coexistent "output" lJ effects E al and E 2a defined by (6.43), then, Eqs. (6.45) and (6.20)
133
imply
(6.42),
so that the apparatus
correlations
described
this apparatus effects F
We
by
indeed measures F 1 and F2, with
FI2 ' FI2,,
etc.
As
remarked
previously,
then also measures - at least implicitly - all other
c B.
shall assume
are d e c i s i o n
that,
effects.
as already suggested by the notation, Then ( 6 . 4 4 )
means
that
the
corresponding
the E?. ij projeca
tion operators project onto four mutually orthogonal subspaces EijHa, and H a is the direct sum of the latter. Moreover, a a commuting projection
operators
a
E 1 a n d E2,
aa
a n d we g e t
aa,
a
E l 2 = E1E 2 ,
(6.43) defines two
E l 2 , = E1E 2
,
etc.,
in accordance with our previous results
(see 6) and with "conventio-
nal"
operations
quantum
mechanics.
Choosing
four
~ij (i = 1 or I',
j = 2 or 2') with ~jl
but
arbitrary
otherwise,
= Fij
we replace
(6.46)
(6.45) by the stronger
require-
ments a
(6.47)
~ijW = Tra((l @ Eij)S(W @ Wa)S*) .
(By
taking
virtue
of
the
trace,
(6.46).
(6.47)
Being
is
analogous
easily to
seen
(5.14),
to
imply
(6.47)
(6.45),
means
that
by the
operation ~ij is performed by selecting those microsystems which have triggered
the
effect
E~. ij
at
the
apparatus
-
i.e.,
the
correlation
effect Fij.)
We are
thus left with
the problem of representing
tions ~ij satisfying, by ( 6 . 4 6 ) a n d
(~2 + ~2' in the form
(6.47),
+ ~'2
four given opera-
(6.21),
+ ~ ' 2 ')I = 1 ,
(6.48)
with suitable Wa, S, and four projection opera-
134 a
tors
E.. satisfying (6.44). An analogous problem has already been ij solved in §5 i n t h e p r o o f of Theorem 2. There two g i v e n o p e r a t i o n s ~
and ~' satisfying (¢* + ¢'*)I = 1 ,
in analogy to (6.48), were represented in a form analogous to (6.47) (cf. (5.20) and (5.21)),
~(')W = Tr a ((I ~ E('))S(W @ Wa)S*) a ' with two projection operators E a and E'a which satisfy the condition E a + E ' a = ia analogous cit
to (6.44). As can be easily seen by inspection,
construction
generalized
of
Ha,
immediately
Wa,
_S, E a
and
E'a described
to the present problem.
in
the expli§5
can
This establishes
be the
existence of the desired quantum mechanical model.
The model as
apparatus
already
remarked,
can also be "used" the operations
to perform operations;
~ij'
by
selecting
e.g.,
the microsy-
stems which have triggered the correlation effects Fij; or the operations ~i = ~12 + ~12' by selecting
and
~2 = ~12 + ~I'2 '
the microsystems which have triggered the effects F I or
F2, respectively; or the non-selective operation
= ~12 + ~12' + ~I'2 + ~i'2' For a given pair of coexistent effects FI, F 2 and given correlation effects
Fij , the choice
still highly arbitrary. tions like ~I' ~2 and ~.
of the four operations The same,
therefore,
~ij
(cf.
(6.46))
is
is true also for opera-
135
The
coexistence
arbitrary
criterion
(6.16)
sets of effects,
A set C c L(H)
may
be
generalized
immediately
to
as follows:
is coexistent
if and only if there
exist a Boolean algebra B and a mapping F
satisfying
: a
(6.10)
B
c
and
+
F
(6.49)
e L(H)
(6.12),
such
that C is con-
flained in the range B = FB = {Fa I a e 2} of F.
Namely,
if C is coexistent,
measurement lar
case
output
there
of all effects
outputs
a
effects
F
form
until a
Conversely, existence
if
apparatus
one
Boolean
c FB ~ C.
algebra
is
arrives
B,
of such an apparatus
b,
above
at
an
and which
(6.49)
satisfied,
c for the joint for the particu-
c can be extended
finally
Therefore
(6.49)
an apparatus
F ~ C. As described
C = {FI,F2} , this
channels,
exists
is we
apparatus
measures
necessary may
by adding
(6.49)
all
coexistence.
consistently
thus taking
b whose
together
for
new
assume
the
also as a suffi-
cient condition.
If
taken
and
the
since here
literally, Boolean
algebra
apparently can have
finitely
of
outputs
of
this
outputs
discussed
bra
As
we
physically
the
With
C,
apparatus before see
condition
called
-
as
in
then to
v
of
(6.49).
a
-
effects
particular
B
' to
the
Boolean
measuring
finitely
case
also
case
also
many
of
"output"
two
alge-
simple
and
infinite
(6.49).
B = FB ~ C also is
C
elements,
together
criterion
set
the successive
the
are
the
type considered
and
there
of
the
in
if
many
only,
as
finite
however,
Thus
c of
and
leads
only
finitely
channels
the coexistence
set
the
of
apparatus
A,
below,
to apply
consist
output
also
seem
possibilities
the
coexistence
of C.
both
c -
satisfying
obviously,
completion
many
interesting
here
B
operations
shall
sets of effects
arguments
a real measuring
application
2"
these
satisfies
coexistent,
C = {FI,F2}
-
a
and
the is
coexistent
136
An arbitrary satisfies since
pair
of effects
condition
{FI,F2}
(6.24),
{FI,F2} ~ C ~ ~_,
and
by
chosen from a coexistent
is
(6.49),
therefore there
are
coexistent. two
set
Indeed,
elements
of
B
1 and 2, say - which are mapped by F into F 1 and F2, respectively. a Boolean algebra,
B contains
ments
(6.6) satisfying Eqs.
etc.
defined
-
i.e.,
by
condition
(6.10)
and
(6.49)
implies
the
(6.19),
coexistence
we
obtain
of
the
mapping
(6.24)
for
the particular
(6.16)
condition
equivalent
then, -
hand,
from •(6.17), F.
trivially (6.49)
coexistence
i and 2 also
As
the four ele-
(6.17) and (6.18). With effects F12 , F12 ,
(6.24)
(6.12)
other
along with
C
is
This
really (6.16)
(6.20)
(6.18)
and
argument
and
the
(6.49)
also
in
shows
(6.24)
that
Since,
this
a generalization and
(6.21)
properties
case C = {FI,F2}.
implies
conditions
Eqs.
case, of
the
the
for pairs
on
two
of ef-
fects.
We do not know whether, provided of
conversely,
a set C of effects is coexistent,
this is true for all pairs {FI,F2} from C. But if C consists
decision
effects
only,
this
can
indeed
be
proved.
Or,
in other
words: A set C of decision effects is coexistent if and only if every pair {EI,E2}
from C is coexistent - i.e., according t o 6: if and only
if C consists of pairwise commuting projection operators.
The
"only if" part of this
der,
therefore,
operators. A,
V
-
(6.32).
stions, limit
of i.e.,
commuting
to these operators
commuting
a Boolean
algebra - with
generated
which
projection
the operations
like,
again
E l... E n E B
and
the
projection
operators,
respect
to these
which
is
operations
B is the unique Boolean algebra of projection
by C.
would
elements
introduce
applying
mutually
In other words,
operators
set C of mutually
Consi-
' as defined by (6.32), we obtain from C a set B ~ C, also
consisting closed
an arbitrary
By successively
and
statement has already been proved.
arise e.g.,
(We leave aside here if B were assumed
some topological
que-
to contain also certain
n÷=limEI A . . . A En with E l, E 2 ... g B.)
notation
EI~¢~...~
E.l A E.3 ~ E.E.1 3 = 0
vious generalization of (6.4) yields
for
En
i @ j.
for For
ElY such
We
...V
En
if
El,
an
ob-
137
E 1A
(E 2 V . . . V E n) = (E 1 A
E 2) V . . . V
(E 1 A
E n) = 0 ,
so that, by (6.32), EI V
(E2 V
...V
E n) = E 1 + ( E 2 V
...V
E n) .
Repeating this argument, we thus obtain
E1 ~
In order we
now
to prove
identify
...~
E n = E l + ... + E n
that C satisfies
the
(abstract)
(6.50)
the coexistence
Boolean
algebra B
condition occuring
(6.49),
in (6.49)
with the above Boolean algebra B of projection operators generated by C,
and
take
for F
the
identity mapping.
definition.
Moreover,
since
in B
represented
by the operators
Finally,
tivity
is
property
(6.12)
the abstract
1 and 0,
is trivially satisfied.
Then FB E B contains elements
respectively,
I and
condition
C,
by
~ are (6.10)
for the identity mapping F the addi-
identical
with
(6.50).
Thus,
indeed,
the
set C is coexistent.
Consider,
more generally,
a coexistent
subset C
of decision effects,
set C of effects
containing a
a Boolean algebra B and a mapping F as
O
in
(6.49).
Since FB = B
B' of B consisting effects,
F
Boolean
F
Co,
there
exists
a nonempty
subset
of those elements a which are mapped into decision
= E . With arbitrary ~ and B s B'
E B are coexistent a
contains
- i.e.,
algebra,
commuting
the
set
the two effects E
projection
B = FB
operators.
contains
also
and
Since B is the
B = "E~ and EB" , F v B = "E~ or EB" and F , = "not E~".
effects According
to !, the latter are again decision effects, and are given by
E E~V
=EE
(6.32)
and
AE
B = E~ + E B - E E B E , = 1 - E
(cf.
=E
(6.33)).
, = E~V
I
EB
(6.51)
= E'
Therefore
B'
contains
along
with
~
and
138
also
~ A ~, ~ V B and
Moreover,
by
(6.51),
coexistent
decision
is closed,
and
tions
A,
mapping
V
~'"' i . e . ,
and
F to B'
effects,
above,
As
algebra,
cular,
C
contains
coexistent jection B
B'
operators
itself
(6.32),
with and
B.
--
of
projection
operators
respect
to the opera-
the
restriction
of
-
the
of B' onto B'. As in the particular
contains
leads
A, V
and
along
with
B of C contains
to the familiar physi' for arbitrary
projec-
generated
is a possible
C
the Boolean
algebra
of proo as shown before,
by C. On the other hand,
choice
for B; hence
B
O
B
is the minimal
coexi-
O
stent completion
The concept
of a given coexistent
of an observable
tions
of
notion
appears
quantum
presenting
discussion
with
defined
operationally
here
An easily which
a
after
the application
very
few
-
such
simple
if
in
terms
indicates
any
-
of
of
since
by
coexi-
the
in §I, observables
are
apparatuses
measuring
apparatus,
an appara-
specified
be an apparatus
"measured
mechanical
thus we
this
by the "clas-
and application.
the apparatus
quantum
apparatuses;
the
especially
or an effect
would
here,
rSle in this connection.
is also completely
example
presented
discussed
of its construction
visualizable pointer
ideas
in some formula-
one. We shall conclude
of observables,
instrument
an observable
sical" description
approach
as a derived
the general
a preparing
tus measuring
the
play an important
In accordance
Like
In
on the contrary
a short
set C of decision effects.
is taken as fundamental
mechanics.
stent sets of effects
them.
of
the whole Boolean o operators generated by C . If thus, in partio decision effects (i.e., C = Co), then every
only
completion
subalgebra
of B' under F - consisting
this homomorphism
of projection
o
by
Boolean
commuting
of the operations
in B'.
Boolean
a
algebra,
is a homomorphism
tion operators
algebra B
i.e.,
' defined
cal interpretation
is
the image B' = FB'
is thus a Boolean
case 6 discussed
a
B'
o
shall
value"
with a scale,
of
the
to a microsystem. observables only
assume
are here
on
observable In practice measured
by
that somehow
139
each
application
yields the
of
the measuring
a well-defined
errors
real number
connected
these measured
apparatus
with
values
any
real
measurement, infinite
idealization.
We leave aside here
idealizations
of
of observables
would become very cumbersome
By
definition,
possible them
measured
it may
values, fied
tageous
also
here,
coordinates
case
are
fore represent
Consider for
of
not
measured measured
"unbounded" components
Borel
sets
on
single
a
no measurement operator
(see (cf.,
F . We
thereby
represents
value
in the Set ~, when
state
W.
in
line
Ch.
like,
the
universal
e.g.,
whose
and which
question
operators thus
of
an
intervals;
more
general
observable
wether
or
F
plete quantum mechanical
e
specify W,
description
and
a
given
into
probability
of
~ -
the in
or,
a yes-
this
the
in other
of the observable.
a
measured
is measured
sense
by an
By definition
finding
"statistics"
the
observable,
L(H).
considered
for arbitrary
states
for
sets
the observable
completely
arbitrary
obtain,
is
not
and is thus to be described
Borel
the
there-
observables.
the
If
position
possible meas-
I, e.g.,
here
I).
a
in the subsequent
interval,
choose
is advan-
I as
include
line
It
system,
tr(F W)
observable
line.
on the given
of
-
thus be identi-
in a given Borel set ~ constitutes
F : ~ + F
F
occur
(i.e., bounded)
[13],
all
besides
is contained
mapping
mapping
e.g.,
contains
as measured
real
the real
microsystem,
certain
The
also
we
description
never
of a particle,
below)
of justifying
but
observables
~ of
an
considered,
real
to a finite
subsets
reasons
value
F ,
we
problem
and could
on the
the whole
Thereby
confined
now suitable
of
effect
to use
that
is clearly
apparatus
which
is finite,
interval
assumption
or even impossible.
observable
numbers
the
In view of
a mathematical
measuring
limiting cases of "real"
technical
class
finite
or momentum
values
the
scale
observables.
the
a
real
every
however,
for all
of
contain
a suitable
discussion
ured
values
of
which
microsystem
value".
precision
the difficult
without
"scale"
in practice
with
scale
the
kind,
single
as the "measured
are given with
this
to a
of
in the
words, the
provide
the
given a
com-
140
In particular,
since
the measured
value will
always
lie on the real
line I and never in the empty set ~, we immediately get
FI = 1 ,
Moreover,
clearly,
measured joint
values
the
in
probability
the
union
sets al... a n (i.e.,
corresponding
F~ = 0 .
finding,
kJi e i of
~ir~j
probabilities
of
(6.52)
in a given
finitely
many
state
mutually
W,
dis-
= ~ for i # j), is the sum of the
for these sets ~i''" an separately;
tr(F~J~iW) = ~i tr(F~iW)
i.e.,
.
(6.53)
i Being true for arbitrary W, this implies the additivity property
F~i
~. F~. 1 i
i of
the mapping
F : ~ ÷ F . In
analogy
to our previous
as
~i"
~i
~i F~ i' here
We
have
thereby
sequences
also
(6.54) we have written,
the "disjoint
omitted that
and
an
Eqs.
First,
disjoint
Borel
upper
limit
(6.53)
and
sets ~i"
in
~i
(6.54)
a union of countably many (arbitrary,
Borel sets ~i is again a Borel set;
hand
(6.53)
of
n
and
(6.54)
exist
also
~i
are
and
assumed infinite
This deserves
rily disjoint) sides
in
union" of the sets ~i
not only for finite but also for countably
of mutually
explanation.
notation,
indicating
to be valid
(6.53)
(6.54)
a little
not necessa-
therefore the left
for infinitely
many
sets
~.. Moreover, in this case, the right hand side of (6.54) exists as i an ultraweak limit. To show this, consider the finite disjoint unions " ~n = ~Ji
"finite"
increase from
operator (6.54). hand
version
with
above
for which
by
n. 1,
in L(H), Then,
side
of
of
Since, they which
we already (6.54). as
elements
indeed
FSn= ~i~n Fai'
nonnegative of L(H),
converge
is also
of ultraweak
convergent,
and
according
operators
they are
ultraweakly
is taken as defining
by the definition (6.53)
The
have
for
FSn
also
convergence, (6.54)
thus
bounded
n + ~
the right hand
to
to
an
side of
the
right
is equivalent
to
141
(6.53)
-
joint
with
unions.
tr(F iW) Wi
arbitrary
latter sent
Finally,
as
='l is
a
finite the
in
or
sented as ~ y
leads
an
argument
detail
is well
is
here.)
not
-
algebra ~
~,
2,
if
e.g.,
already
only
defines as
(6.3))
indicated
finite
our
Borel
sets
is
o-algebra.
The
mapping
(6.12).
An a
observable,
therefore,
particular
(6.24), the
with
real
line
standing I,
is quite
natural
This
description
is
of
coexistence
observable Eqs.
since,
measures
B
~,
as
and
in
not be
the pre-
repre-
I form the
a Boolean
intersection
"not a" = ~'
as
of this Boolean
The
fact
that
B
intersections
expressed
by
and
calling
(6.24),
represents
described
a
above
since Eqs.
quantum
for
the
Boolean
o-algebra additivity
o-algebra
in accordance a
the
with
measuring
all effects
F
is
of
of
Borel
the
e FB,
set.
for
as
sets
on
(6.54)
-
of
operational
apparatus
of
condition
property
- in place
a
(6.52)
mechanically
sense
of
Boolean
a generalization
in
a generalized
not
unions
2
considered
effects
indeed,
alge-
contains
I a e 2} = B is thus a coexistent
be
the
real line I and the empty set ~,
of
together
shall
B ~ ~ may
line
elements
infinite
for a Boolean
also
(Although we
set
and with
which
real
a~
(6.54)
may
coexistent 2
on the
in condition
whereas {F
whether
(6.55)
F : B ÷ FB = B c L(H)
The set of effects
of
~ _c B •
union
usually
considered
(6.10),
union.
g with
dis-
of t r ( F ~ i ~ W)x and
trivial,
set
notation.
arbitrary
to
a
infinite
regardless
disjoint
"trivial"
countably
correspond
(6.53),
if
the
also
of the kind
to
"~ and B" = ~ A
but
mapping
interpretation
are the whole by
countably
set ~, (6.54) also implies
sets
of the set a. The
(cf.,
as
one
Borel
"~ or 8" = a v B
complement bra
the
for
entirely
Since
with a suitable
known,
also
infinite
F~ < F~
As
-
the physical
probabilities
case it
W ~ K(H)
(6.12). meaning
the
as explained
given above.
(6.52) and (6.54) also imply the relations
F,
=
I - F
=
F'
(6.56)
142
and
F
~v
= F
B
+ FB - F
~
r~B
(6.57)
"
(Compare the d e r i v a t i o n of (6.13) and (6.14) from (6.10) and (6.12).)
A mapping F
: ~ ÷ F
tions
(6.52)
-
more
or,
ted by
and
of Borel
(6.54)
briefly:
into
called
a
effects
positive
satisfying
of an
~ : ~ + m(~)
of
"ordinary"
Borel
sets
(normalized) ~ into
the condi-
operator-valued
a POV m e a s u r e - on the real line.
the d e f i n i t i o n
a mapping
is
sets
This
is motiva-
measure,
nonnegative
measure
w h i c h as
numbers
m(u)
is c h a r a c t e r i z e d by the a n a l o g o u s conditions
~(I)
(In
particular,
and
(6.53),
= 1 ,
these
m(¢)
= 0
conditions
,
m ( k ? ~ i) = ~ m ( ~ i ) • i i
are
satisfied,
by
a POV measure
F
for ~(a) = tr(F W) with
virtue
of
(6.52)
and an a r b i t r a r y
state W.)
If only p r o j e c t i o n operators urements,
as
occuring fects, sets
in
F
in " c o n v e n t i o n a l " the
above
decision
(6.52) and
one
must
arrives
effects
as d e s c r i b i n g
quantum mechanics,
discussion
= E . Thereby
into
are considered
be
at
assumed
a mapping
(projection
yes-no meas-
then all to E
operators),
be
effects
decision
F ef-
: ~ ÷ E
of Borel
which
satisfies
(6.54) in the form
E1 = 1 ,
E~ = 0
(6.58)
and
E kLy (x. = ~. E . ' I i i i and
is
called
mechanics, ure E
on
a
spectral
therefore, the
real
measure
on
every o b s e r v a b l e
line,
whereas
I.
(6.59)
In
"conventional"
corresponds
quantum
to a spectral meas-
in the f r a m e w o r k c o n s i d e r e d here such
observables - the so called d e c i s i o n observables - c o n s t i t u t e a particular
class
only.
Since
the
decision
effects
E , a c B
form a c o e x i -
143
stent set,
they are represented by mutually commuting projection
ope-
rators, and according to (6.51) we also have
E
(The
last-mentioned
(6.59),
by using
~8
properties
E E8 .
=
can
the elementary
(6.60)
also
result
be
derived
directly
from
that the sum E + E of projec-
tion operators E and E is again a projection
operator if and only if
EE = EE = 0.
implies
With
for
arbitrary
B,
Eq. (6.59)
(6.59),
disjoint and
= (~t'~B) ~ ( a ' ~
Borel
the 8)
EB = E
t.~ ~ + E , ~ B .
joint,
this
trivially
this
immediately
sets y and 6. For arbitrary
decompositions lead
Because
finally
implies
generalized
to
EyE~ = E6E Y = 0
to the
a = (~f'~ B ) ~ E
sets
= E
a ~B'
E E B = EBE ~ = E
finite
(a f'~B')
and
B,
and
r~B + E and
~B.)
intersections
sets ~ and
a'~
B
are
Eq. (6.60)
~i t'~''" q'~=n'
dis-
may but
be ac-
tually it is true, in the form
E
=
~
E
i arbitrary
sets
a.. In the latter case, the right hand side of (6.61) exists, i has to be understood, as the ultraweak limit of ~i~n E~ i for
n + -.
We
will
(6.56)
and
not
(6.57)
or
(6.61)
for
and
finite
,
1
prove
are also
countably
this
infinite
here.
intersections
Besides
(6.61),
of
Borel
clearly,
true for spectral measures,
Eqs.
i.e., with E
for F . (Compare also (6.51).)
The more
familiar description of decision observables
operators projection
is
obtained
operators
as E(%)
follows. by
Define
inserting
a for
by self-adjoint
one-parameter ~
the
family
intervals
of
(-~,%]
into the spectral measure E : ~ ÷ E ; i.e., E(k) = E(_=,k]
By
(6.55),
(6.58)
characteristic
and
(6.59),
then,
E(k)
•
(6.62)
may
be
shown
to have
properties of a spectral family of a self-adjoint
rator A, namely:
the ope-
144
i) %1 < %2 implies E(% I) < E(~2). ii) For
% ÷ -=
and
respectively. confined
% + +~,
(For a
E(%)
converges
"bounded"
to a finite
interval
ultraweakly
observable,
to 0 and
with measured
I,
values
[Al, A2] , we simply have E(%) = 0
for % < A I and E(%) = i for % ~ A2.) iii) For sequences ultraweakly
%i ~ % converging
to E(%),
from below,
whereas
to % from above,
E(~i)
converges
%. $ % converging to i also has an ultraweak limit, which is < E(%)
E(%i)
for sequences
(by i)) but need not be equal to E(%).
According every
to the
such
famous
spectral
spectral
family
E(%)
theorem
for self-adjoint
uniquely
determines
a
operators,
self-adjoint
operator A = ~ % dE(%) ;
conversely,
every self-adjoint
form
with a unique
(6.63)
i) to iii) requiring
above.
tral
red
(6.63)
is
including, (cf.,
bounded,
family E(~)
than ultraweak
e.g.,
e.g.,
(6.63)
A may
be
represented
satisfying
ii) and iii) are usually
We will not present
theorem
in Eq.
spectral
(Conditions
strong rather
difference.
operator
(6.63)
convergence,
[13], defines
conditions
formulated
by
but this makes no
here a rigorous version a rigorous
in the
the
integral
Ch. 4). - If the observable
conside-
a
definition
bounded
of
of the spec-
self-adjoint
operator
A.
Such operators have been called Hermitean in the preceding chapters.)
Usually the
a quantum
operator
vable
A".
description
mechanical
A as given
We
need
not
(decision)
by (6.63), rederive
such as, e.g.,
here
observable
and is simply
is represented called
the well-known
by
"the obser-
features
of this
the connection between the possible measur-
ed values of the observable and the spectrum of A, or the formulae
W = tr(AW) and
(6.64)
145
(AwA)2 = tr(A2W) - (tr(AW)) 2
for
the
expectation
an observable
the
latter.
observable
considered
the mean
also
-
Eqs.
spectral
and
the
family
or,
spectral
is
measure
from A - because,
Eqs.
(6.68)
AwA
of
from the defini-
and
(6.69)
below.)
the
corresponding
If
the
operator
and (6.65) make sense for states W in a
uniquely
E : a ÷ E
of this point see [14].
determined
may
be
by
the
=
operator
reconstructed
in terms of the characteristic
xa(k) of the Borel set ~, E
deviation
E and the physical meaning
equivalently,
(6.64)
E(k)
square
they follow immediately
"domain" only. For a discussion
The
i.e.,
and
of the spectral measure
(Compare
A - is unbounded, suitable
W
A in a state W;
tion of A in terms of
value
(6.65)
I
for k ~
0
otherwise
A,
from E(k)
-
function
is given by Cl
= f
E
The description is
therefore
measure
E,
of a decision
completely
and
it
vant quantities
The
preceding
Xa(k) dE(h) = X (A) •
is
observable
equivalent
also
vable,
as decribed
called
"the observable
the
practically
can be calculated
construction
to
b y a self-adjoint
useful
directly
the
expectation
observable
F
spectral
physically
rele-
obser-
F : e + F , and therefore
simply
F" in the following.
to an
In analogy
to (6.62),
one
spectral family"
completely
analogous
value
and
W
in a given
a
arbitrary
F(X) = F(_,,X]
properties
since
by
in terms of A.
can be generalized
by a P0V measure
may define a "generalized
with
description
operator A
state
to
,
i)
the mean W may
then
(6.66)
-
iii)
square be
above.
deviation
calculated
From AWF
as
F(k), of
the
Stieltjes
146
integrals with the weight function m(X) = tr(F(X)W)
(6.67)
w : f % dm(X)
(6,68)
according to the formulae
and (AWF) 2 : f (l - W )2 dm(X) : f 12 dm(l) - (w)2
These formulae follow easily from the probability
dm(l) = m(l+dl) - m(X) = tr((F(X+dX)-F(X))W~
(6.69)
.
interpretation
of
= tr(F(l,l+dl]W ) .
In analogy to (6.63), one may now attempt to define an operator
A = f i dF(l)
in
terms
could
of which
then
(6.64)). when
rewritten
even
this
rewritten for
a
square
via
whereas
fX2dE(X) = A 2, and imply
spectral A
defined
ferent an
neither family by
thus
(6.70),
A
to
it
form
a
not
directly
a
decision
side
of
need
a given
with
a
can
(6.65)),
to
bounded
be
also
(6.70)
equal
be
so that
calculate
(6.63) Eq.
not
mean
operator implies does not
to tr(A2W>.
nor even the generalized uniquely
operator
of the form (6.70).
associated
not
F : = ÷ F
can be reconstructed since
for all W. But
from the corresponding observable
[jl2dm(X)
(cf.
self-adjoint
(6.69)
possible
fX2d(tr(E(1)W~ = tr(A2W), that
(6.68)
tr(AW)
bounded
W = tr(AW)
hand
is
values
(6.70) may indeed be shown -
define
that
right
the POV measure
representations
operator
F
(6.65) for
F(I)
familiar
form tr(A2W) - (tr(AW)) 2 (cf.
/12dF(X) = A 2 , so
Moreover,
-
(6.70)
the expection
more
such
the
observable
deviations
Indeed,
A,
case
in the usual
-
F is bounded,
operator
simple
general
the
appropriately
Hermitean) in
formally
in
If the observable
interpreted
(i.e.,
A.
be
- at least
,
from
A may have
(For instance, observable
F
the operator several
dif-
the Hermitehas,
besides
147
(6.70),
at
the
provided
one
least
one
additional
by
the
representation
spectral
theorem,
tral family E(%) of A in place of F(%).) is
unbounded,
"domain
the
definition
problems"
similar
ones
associated
with
even
be densely
defined,
tion may
consist
such difficulties
to,
Eq.
of
but
actually
zero vector
form,
namely
the "ordinary"
the
more
in extreme
are, however,
with
this
spec-
If the observable considered
of
even
(6.63); and
the
(6.70)
of
operator severe
the
leads
than,
operator
cases
only.
A
the known A
its domain
(Observables
rather "pathological"
to
need
not
of defini-
F leading
to
from the physi-
cal point Of view as well.)
But
even
defined,
if - as
from
adjoint
a bounded
obervable
- the operator A is well-
it provides a rather incomplete description of the correspon-
ding observable, ted
for
it,
since not even mean square deviations can be calcula-
and
operator
observable,
therefore
it
associated
therefore,
has
is
with to
much a
be
less
decision
described
useful
than
observable. either
the A
self-
general
directly
by
the
corresponding POV measure F, or alternatively by the generalized spectral
family F(%)
cal interest
associated with
it,
from which
quantities
of physi-
like W and AWF can also be calculated via Eqs.
(6.67)
to (6.69).
Although here, be
quite natural
the consideration
merely
concrete
of
academic
physical
in the version
of quantum mechanics
of such more general interest
examples.
if
Such
an
it
observables
could
not
application
be -
in
presented
would
still
illustrated fact
the
by only
really interesting one known up to now - concerns the localization of massless
particles.
Since
this
subject
has
been
discussed
in
much /
detail elsewhere [15], we shall sketch here the basic facts only.
A position observable, tum mechanical more
general
particle
type
than
describing
the spatial
at a fixed time, the
observables
localization of a quan-
t = 0 say, considered
is of a slightly so far,
since
its
"measured values" are the three spatial coordinates
of the particle -
i.e.,
ones.
triples
of
real
numbers,
rather
than
single
Therefore
a
148
position
observable
defined
on
than
the
on
effect
the
Borel
real
F is
is
to
be
sets
line
realized
~
at
idealization, val,
rather
ure F
An
elementary
rized
by
genous ble
its
particle
unitary
and
(6.54),
covariance of
the Euclidean
Conventional
quantum
E ,
and
particle the
covariant
of P.
mechanics counters
corresponding
spectral
f
R J,
rather
measuring
and sufficiently
the
simple
the spatial volume ~ and again
sensitive
in
and
is thus
represents
some
time
expected
an
inter-
to meas-
measure
quantum under
mechanics Poincar~
for the position obser-
has
only
to
satisfy
to be described
on
From
such
to
(6.52)
a certain
representation
observables.
by decision becomes
a
state
are called Euclide-
decision
E : ~ + E
its
analogous
Such POV measures
admits
on
irreduci-
condition
conditions
P
inhomo-
group
measure
mapping
(i.e.,
by a suitable
to the given unitary
are
is characte-
Poincar~
the
POV
respect
subgroup
in [15].
cordingly,
the
besides
with
an covariant
apparatus
as described
of
corresponding
condition
F : ~ ÷ F
space
condition
properties
to an additional
particles:
the
is
in relativistic
representation
of such
measure
at best.)
transformations,
space H. This leads vables
last
time only,
transformation
Lorentz)
effect
counter
than at a sharp
POV
- for finite
(The
real
only approximately
a
counter occupying
t = O. a
by
three-dimensional
An
physically
time since
in
I = R I.
at least - by a particle "operating"
described
an
spectral
Ac-
effects
Euclidean measure,
a
{xl)
position operator
X
=
X2 X3
with
three mutually
structed, Wigner
and
[16]
vice
- and
the above-mentioned sition
operator
X
commuting versa. later
self-adjoint
It has on, more
requirements for
massive
been
components
shown
rigorously,
long
by Wightman
lead to a unique particles
and
X. can be coni ago by Newton and
for
[17]
- that
"Newton-Wigner" massless
po-
particles
149
with
spin
zero,
whereas
for massless
particles
with spin they can not
be satisfied at all.
Massless appear nics, by
particles
to be not whereas
suitable
ved tral
localizable
counters.
measures at
spin
in practice
by admitting
arrives
with
This
a quite
as
e.g.,
according
at least
Euclidean
only,
-
satisfactory
for elementary
particles
the interested
reader is referred
neutrinos
can
POV measures,
rather
observables.
description
of position
mass
and spin.
to the literature
thus
be localized
indeed
position
of arbitrary
-
quantum mecha-
can certainly
discrepancy
covariant
or
to conventional
photons
apparent
describing
photons
be
resol-
than Thereby
specone
measurements
For more details
([15],
[7]).
150
References
[i] Kraus,
K.: Operations and Effects
in the Hilbert
Space Formula-
tion of Quantum Theory. In:
Hartk~mper,
Mechanics
A.,
Neumann,
and Ordered Linear
H.
(Eds.):
Spaces.
Foundations
of Quantum
(Lecture Notes in Physics,
Vol 29.) Springer, Berlin / Heidelberg / New York, 1974; p. 206 - 229. [2] For
the most
recent
account
of
Ludwig's
theory,
and
for many
additional references, see: Ludwig, G.: Foundations of Quantum Mechanics I. Springer, New York / Heidelberg / Berlin, 1983. [3] Davies, E.B.: Quantum Theory of Open Systems. Academic Press, London / New York / San Francisco, 1976. [4] Schatten, R.: Norm Ideals of Completely Continuous Operators. Springer, B e r l i n / G~ttingen / Heidelberg, 1960. [5] Haag, R., and Kastler, D., J. Math. Phys. 5, 848 - 861, 1964. [6] Stinespring, W.F., Proc. Amer. Math. Soc. 6, 211 - 216, 1955. [7] Kraus, K., Ann. Phys. (N.Y.) 64, 311 - 335, 1971. [8] Gorini,
V.,
Kossakowski,
A.,
and
Sudarshan,
E.C.G.,
J.
Math.
Phys. 1/7, 821 - 825, 1976. [9] Lindblad, G., Commun. Math. Phys. 48, 119 - 130, 1976; 65, 281 294, 1979. [I0] Naimark, M.A.: Normed Algebras. Wolters-Noordhoff, Groningen, 1972; §22, No. 3.
151
[11] Wigner,
E.P.:
Group
Theory
and
its Application
to
the Quantum
Mechanics of Atomic Spectra. Academic Press, New York / London, 1959; Ch. 26. [12] Hellwig,
K.-E.,
and
Kraus,
K.,
Commun.
Math.
Phys.
II,
214
-
220, 1969; 16, 142 - 147, 1970. [13] Jauch, J.M.: Foundations of Quantum Mechanics. Addison-Wesley, Reading (Mass.), 1968. [14] Kraus,
K.,
and
Schr~ter,
J.,
Intern.
J. Theor.
Phys. 7,
431 -
Uncertainty
Prin-
442, 1973. [15] Kraus, K.: Position Observables of the Photon. In:
Pryce,
W.C.,
Chissick,
S.S.
(Eds.):
The
ciple and Foundations of Quantum Mechanics. Wiley, London / New York / Sidney / Toronto, 1977; p. 293 - 320. [16] Newton,
T.D.,
and Wigner,
E.P., Revs. Mod. Phys. 21, 400 - 406,
1949. [17] Wightman, A.S., Revs. Mod. Phys. 34, 845 - 872, 1962.
Lecture Notes in Physics Vol. 152: Physics of Narrow Gap Semiconductors. Proceedings, 1981. Edited by E. Gernik, H. Heinrich and L. Palmetshofer. XIII, 485 pages. 1982. Vol. 153: Mathematical Problems in Theoretical Physics. Proceedings, 1981. Edi{ed by R. Schrader, R. Seiler, and D.A. Uhlenbrock. XII, 429 pages. 1982. VoL 154: Macroscopic Properties of Disordered Media. Proceedings, 1981. Edited by R. Burridge, S. Childress, and G. Papanicolaou. VII, 307 pages. 1982. Vol. t55: Quantum Optics. Proceedings, 1981. Edited by C.A. Engelbrecht. VIII, 329 pages. 1982. VoL 156: Resonances in Heavy Ion Reactions. Proceedings, t981. Edited by K.A. Eberhard. XII, 448 pages. 1982. Vol. 157: P. Niyogi, Integral Equation Method in Transonic Flow. Xl, 189 pages, t982. Vol'. 158: Dynamics of Nuclear Fission and Related Collective Phenomena. Proceedings, t981. Edited by P. David, T. Mayer-Kuckuk, and A. van der Woude. X, 462 pages, t982.
Vol. 173: Stochastic Processes in Quantum Theory and Statistical Physics. Proceedings, 1981. Edited by S. Albeverio, Ph. Combe, and M. Sirugue-Ooilin. VIII, 337 pages. 1982. VoL 174: A. Kadie, D.G.B. Edelen, A Gauge Theory of Dislocations and Disclinations. Vtl, 290 pages. 1983. Vol. 175: Defect Complexes in Semiconductor Structures. P roceedings, 1982. Edited by J. Giber, E Beleznay,J. C. Sz6p, and J. L~.szI6.Vl, 308 pages. 1983. VoL 176: Gauge Theory and Gravitation. Proceedings, 1982. Edited by K. Kikkawa, N. Nakanishi, and H. Nariai. X, 316 pages. 1983. VoL 177: Application of High Magnetic Fields in Semiconductor Physics. Proceedings, 1982. Edited by G. Landwehr. XlI, 552 pages. 1983. VoL 178: Detectors in Heavy-lon Reactions. Proceedings, 1982. Edited by W. yon Oertzen. VIII, 258 pages. 1983. VoL 179: Dynamical Systems and Chaos. Proceedings, 1982. Edited by L. Garrido. XlV, 298 pages. 1983.
Vol. 159: E. Seiler, GaugeTheories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics. V, 192 pages. 1982.
Vol. 180: Group Theoretical Methods in Physics. Proceedings, 1982. Edited by M. Serdaro~tu and E. inTn(J. Xl, 569 pages. 1983.
Vot. 160: Unified'Theories of Elementary Particles. Critical Assessment and Prospects. Proceedings, 1981. Edited by P Breitenlohner and H.P. D~rr. VI, 217 pages. 1982. " VoL 161: Interacting Bosons in Nuclei. Proceedings, 1981. Edited by J.S. Dehesa, J.M.G. Gomez, and J. Ros. V, 209 pages. 1982.
Vol. t81: Gauge Theories of the Eighties. Proceedings, 1982. Edited by R, Raitio and J. Lindfors. V, 644 pages. 1983.
VoL 162: Relativistic Action at a Distance: Classical and Quantum Aspects. Proceedings, 1981. Edited by J. Llosa. X, 263 pages. 1982. VoL 163: J.S. Darrozes, C. Francois, M6canique des Fluides Incompressibles. XIX, 459 pages. 1982. VoL 164: Stability of Thermodynamic Systems. Proceedings, t981. Edited by J. Casas-V,~zquez and G. Lebon. VII, 321 pages. 19~2. VoL 16~: i'~. Mukunda, H. van Dam, L.C. Biedenharn, Relativistic Mo~Jelsof Extended Hadrons Obeying a Mass-Spin Trajectory Constraint. Edited by A. BShm and J.D. Dollar& Vl, 163 pages. 1982. Vol. 166: Computer Simulation of Solids. Edited by C.R.A. Catlow and W.C. Maekrodt. Xtt, 320 pages. 1982. Vol. 167: G. Fieck, Symmetry of Polycentric Systems. VI, 137 pages, 1982. Vol. 168: Heavy-Ion Collisions. Proceedings, 1982. Edited by G. Madurga and M. Lozano. Vl, 429 pages. 1982. Vol. 169" K. Sundermeyer, Constrained Dynamics. IV, 318 pages. 1982~ Vol. 170: Eighth International Conference on Numerical Methods in Fluid Dynamics. Proceedings, 1982. Edited by E. Krause. X, 569 pages. 1982. VoL t71: Time-Dependent Hartree-Fock and Beyond. Proceedings, 1982. Edited by K. Goeke and P.-G. Reinhard. VIII, 426 pages. 1982. Vol. 172: Ionic Liquids, Molten Salts and Polyelectrolytes. Proceedings, 1982. Edited by K.-H. Bennemann, F. Brouers, and D. Quitmann. VII, 253 pages. 1982.
VoL 182: Laser Physics, Proceedings, 1983. Edited by J.D. Harvey and D. E Walls. V, 263 pages. 1983. Vol, 183: J.D. Gunton, M. Droz, Introduction to the Theory of Metastable and Unstable States. VI, 140 pages. 1983. Vol. 184: Stochastic Processes - Formalism and Applications. Proceedings, 1982. Edited by G.S. Agarwal and S. Dattagupta. Vl, 324 pages. 1983. Vol. 185: H.N.Shirer, R.Wells, Mathematical Structure of the Singularities at the Transitions between Steady States in Hydrodynamic Systems. XI, 276 pages. 1983. Vol. 186: Critical Phenomena. Proceedings, 1982. Edited by EJ.W. Hahne. VII, 353 pages. 1983. Vol. 187: Density Functional Theory. Edited by J. Keller and J.L. G~.zquez.'4, 801 pages, t983. Vol. 188: A. R Balachandran, G. Marmo, B.-S. Skagerstam, A. Stern, Gauge Symmetries and Fibre Bundles. IV, 140 pages. 1983. VoL 189: Nonlinear Phenomena. Proceedings, 1982. Edited by K. B. Wolf. XII, 453 pages. 1983. Vol. 190: K. Kraus, States, Effects, and Operations. Edited by A. BShm, J.W. Dollard and W.H. Wootters. IX, 151 pages. 1983.
