Introduction to photon communication

,8---"00

= e-

f/(1-E)(1-2Ef/) 2 2 U


Pa

(

0, U TJ VE (1 - €)

)

(1.3.2)

I. Elements of Quantum Mechanics

49

where

4>;,. (u)=Tr (Paexp(-u*a)exp(uat ))

=e-¥ Tr (P'aD(u)) (1.3.3)

u=

Ul +iU2

andD(u) is the displacement operator (see § 1.2.2).

To demonstrate the results, it is first noticed

It is then proved that the operator fit =

2

J 11f(1-f) if , is such that (1.3.4)

which is interpreted as the characteristic function of the operator

~ = yTilii + JI'"=7jVi for

vlO)

=(Vi + iV2) 10)

=

1]

~ 1, where Vi is defined such that

0.

Remark: This result means that the photodetector works as a classical device ("l = 1) provided that a vacuum field is added to the signal field via the beam splitter. Finally, for u E IR, and (1.3.5)

50

Introduction to Photon Communication

eq. (1.3.3) leads to

4> ~ (u)

= exp (u ~e({3)) exp ( _ ~2

(~1 + ~) )

which is typical of a Gaussian random variable X: exp (iu(X) -

u i 2u

(1.3.6) ~X (u )

==

2 ).

1.3.2 Heterodyne Here

Wi =1=

O. The heterodyne measurement operator can be

shown to be

~ a2

= '12 (""a + ~t) ai

(1.3.7)

Details of the calculations are similar to those given in the preceeding subsection. The field to be considered is

c = .JiiCO + vr=r7(V'ie-i(wo+w;)t + 'V2 e - i (WO-Wi )t) CO =

.;e(ae-i(WO+Wi)t + li'ie-i(WO-Wi)t) + .Jf='€ b e- iwot (1.3.8)

Let us apply these results in the limit of

f3

~ 00,

to a spe-

cific example that is very useful in optical communication. Let the density operators of the signal and image fields be given by

1r1 Jexp (_laN~12) la)(ald a 1r1 Jexp ( _1~~2) la)(ald2 a 2

Pa({3) = {

Pi

=

1

(1.3.9a, b)

2

Using the signal-image independence, the operators

(1.3.10a, b)

I. Elements of Quantum Mechanics

51

are commuting. The joint characteristic function is found to be of Gaussian shape

= exp ( u ~e (as) + v SSm (as) )

exp ( _ Nt

+ ~2 + 1 (u

2

2 ;

v

) )

(1.3.11) where u, v E IR. In the case that N 2 ~ N 1 , eq. (1.3.11) is simplified

so that

uh = ~ + ~

which is larger than the value found in the

case of homodyne detection where u~ = ~

+i

(see eq. (1.3.5).

Another measurement scheme has recently been extensively studied because it provides new interesting applications in quantum optical communication. It is called quantum non demolition measurement because the observable is measured without any perturbation. 1.3.3 Quantum Non Demolition

Suppose the process X(t) is observable by the measurement of

Y(r) so that the commutation relation [X(s), yet)] = 0, \:It ~ s without any restriction on the choice of the observables measured at future time instants,

[Yet), feu)] = 0,

u?t

After the measurement at time s, the state of X(t) is conserved at all times. Such a measurement is called of Quantum Non-

Demolition (QND) type [34]. The main consequence is that the back action in the process may be evaded, therefore repeated measurements become possible.

52

Introduction to Photon Communication

Experimental realisations have been reported using a set up of the type of Fig. 0.3. using a lossless Kerr effect characterized by an intercation Hamiltonian

Ii; - X(ws,wp)NsN;.

tions are for Ns the measured observable,

The required condi-

[Ii;, Ns]

= 0 (back-action

= 0 (QND measurement) and the 5; readout observable (sine operator of the probe field) [Ii;, 5;] # 0 [15].

evasion), [Ns(O),N:(t)]

Application of QND to optical communication is briefly examined in an example treated below. Consider a QND transmission scheme where we denote

X

Xl = l(~ + ~ t) of eigenstates IXI) and Y =~ = ii(a;, - a;, t) of eigenstates IX2). The evolution operator U(O, t) of the system (a signal with a probe), characterized by the interaction Hamiltonian

Ii;(B) = AXX: ~,

is given by

U(s, t) = exp ( -ix

it

Ii;(B)dB)

Initially, we suppose that the signal and probe are independent and prepared in squeezed states so that 14>8): ((Xl), u;), l1Jp): ((X2) =

0, u;). At time t, the density operator is given by A(t) = 11f')(1f'I, where I1/;)

= 14>8)I1Jp)·

To evaluate Prob(xl' X2; t) the probability distribution of outcomes

{Xl, X2}, we employ the evolution equation in the interaction picture, where we set demonstrated that

llia) = IXI)(Xll and ll~b) = IX2)(X21. It is easily

I. Elements of Quantum Mechanics

where

53

N x( m, ( 2 ) denotes a Gaussian distribution of the r.v. of

mean m and variance u 2 [33]. The marginal distribution for the probe field is Nx2(2Xt(Xl), (2Xt)2 u;

for Xt --+

00,

+ u;).

Prob(xllx2i t ) --+ N X1 (f:J"

It is also noticed that

f:t)

which means that

after the outcome X2 is registered, the variance of the signal observable is significantly decreased. An interesting situation occurs when the intensity of the probe is large enough. In the case of photon number measurements (the

readout observable becomes

s;, = sin(~)), the measurement accu-

racy is proportional to the variance of the phase of the probe field. Therefore a definition of the phase operator is absolutely required. This question is analyzed in some detail in Sect. IV.

1.4 MeasurelIlent Processes

One of the problems currently discussed in the theory of measurements in quantum mechanics, is related to the wave-packet reduction postulate (WRP). According to this postulate, after a

measurement of an observable of the system by an apparatus, the system evolves from a mixture of states of positive entropy to a pure state of zero entropy. This apparently contradicts the classical thermodynamics laws. Moreover the information transfer from the system to the apparatus is an irreversible process. In fact, to understand the physics of the measurement, the process should be examined on the basis of whether or not the measurement is read. This is considered in the following section.

54

Introduction to Photon Communication

1.4.1 Time Evolution Let a system S be given on which the observable terized by the self-adjoint linear operator on ~(a)

Ek akIlk

~(a)

· We set lak)(akl - Ilk

rt,

A is charac-

such that

A ==

•

wIth the orthogonal resolu-

tion of the identity ~(a)~(a)

IIi

Ilk

~~(a)

L...J Ilk

== 0 == 11

(1.4.1 )

k

Let

A be the density operator of the system at the initial time s.

Consider now this quantum system S coupled with I, an apparatus (or instrument) initially separated from it. This apparatus, utilized to provide information about S, is in interaction with S under the

self-adjoint Hamiltonian H=

Ho+if;I, Ho being the Hamiltonian

of the free system. The system evolves with a unitary operator U(s, t) following the evolution postulate, and obeys (the model is simplified by consid-

ering that all operators act on the same Hilbert space 1i)

d '" dtA(t)

= -Iii ['" H,A"'1J

A(t) = U(s,t)A(s)Ut(s,t) U(s, t)

= exp (

-* J.t

(1.4.2)

H(19)d19)

To simplify the presentation of this section, only a few results of the theory of quantum measurements established by von Neumann are utilized here. A point of view suitable for information communication purposes is adopted here.

I. Elements of Quantum Mechanics

55

For a single measurement and after an interaction at t, S and I are separated. The outcome ak is registered with the probability according to the fundamental postulate (postulate of observables) "...

~(a)

Prob(ak)=Tr[A(t)lIk

]

(1.4.3)

Its average value is given by

After· such an observation, the density of the system becomes by the WRP:

Notice that if the measurement is not registered, the final density IS

~/(t) =

L ~(t) Tr [A(t)1i.;(a)] = L 1i.;(a) A(t)1i.;(a) k

k

Initially in the state I¢(O)) = lak) ® l
(Ii~a»)) on the apparatus means that the system S is in lak)' This type of measurement is called ideal. Now according to the superposition principle if we start with

I¢(O)) = Ek Pklak)®lcp), we end up with I¢(t)) = Ek Pklak)®li~a») for which quantum correlations appear between the macroscopic states lak) ® li~a») and la,) ® li~a»). Moreover, it is possible to set

Ek lak) ® li~a»)

=Em qmlbm) ® li~») which means that the appa-

ratus is measuring a different observable

B of S.

56

Introduction to Photon Communication

Several physicists propose to utilize this ambiguity to encrypt information for secure transmission. A slightly more general concept is perhaps more suitable to define a generalized observable. Let Ci on IC be the initial density operator of the instrument I with a pointer observable interaction with the system of density operator

f,. (a).

A occurs

The

at the

time t. The fundamental postulate, also called the postulate of

reproducibility is ~(a) (~

Tr [(110Ik

~(a)

The set {tItk

~(a)

~(a)

~

t)] = Tr [~(a)] AYik

} need not be orthogonal. Therefore, we choose

them so that Yii Ek'tPklIIk

~

) U(s, t)(A 0 Ci)U(s, t) (t)

lak)

=

VJkilak). Now, from

~(a)~(a)

Ek Yii

Ilk

=

,we have the following properties for the non 1- oper-

ators

Let us extend this formalism to the case of repeated measurements made by two different apparatus I a , Ib on

A and B at

t and t'.

After the measurement ak is registered, the system is in A'(t). At

t', we have A(t') = fi( t, t')A'(t)fi t(t, t'). Therefore, using the WRP, we obtain the conditional measurement probability

I. Elements of Quantum Mechanics

Prob(bqlak)

~

~(b)

= Tr [A(t')llq

57

]

~(a) ~

~(a)

= Tr [D(t,t,)(lh __A(t!!!~

.

) Dt(t,t').u;(b)]

Tr[A(t)lh( )] (I.4.4a) The joint distribution being given by

As demonstrated in [31], if we treat the whole system S + I a

+ Ib,

only the first postulate of quantum mechanics is necessary in order to obtain equations (I.4.4a,b). An important question of current interest can be stated as follows: the apparatus being quantum mechanically described, what is the process leading to the selection of the right p.o.m. among the •

~(a)

varIety Ilk

~(b)

, llq

?

, ....

An answer to this question was recently developed [18,19] suggesting that the apparatus I is actually in interaction with the environment £, via the Hamiltonian

fi Ie .

In case such interaction dominates (the system S that is microscopic is not directly in interaction with £, that is macroscopic) ~

~

and assuming that HIe and He are diagonal in the same normalized basis {lin)}, the final state after measurement can be written

=

L:n cnlsn(t)) ® lin(t)) ® len(t)) for which we notice that there is, at first, a measurement of S by I, and after a QND measurement of I by £. This latter is an ideal one as for t ~ 00, the environment states become uncorrelated (em(t)len(t)) ~ 8mn . as l1f'f(t))

58

Introduction to Photon Communication

Let us briefly see how the entropy functions evolve during these steps. 1.4.2 Entropy Evolution We adopt the conventional interpretation of the concept of entropy in the sense that it is not an observable, i.e. an operator on Hilbert space but rather a property specifying the state of the system described by the density matrix

p.

In fact, a measurement

which is basically a transfer of information is always incomplete and the lack of knowledge from the maximum of available information can be characterized by the entropy. Among the different definitions of the entropy, we select the von Neumann entropy that satisfyes the basic mathematical conditions and the statistical physics requirements

S(p)

= -Tr

(plog p)

(I.4.5a)

Notice that for a system having a given Hamiltonian it, the entropy defined by (I.4.5a) is such that where

:t p =

[p, it].

:t S = -Tr

(it. [p, log jJ]) =

0,

Similarly, the relative entropy is

S(Allp) =

-Tr

p(logp-logA)

(I.4.5b)

whose the properties of positivity, convexity and the usual inequalities can be demonstrated. We just mention here, in a a simplified presentation, a very few of the results which are of direct relevance with the subject of these lecture notes. We refer to the book by Thirring for details and for the proofs of the results mentioned here [17].

I. Elements of Quantum Mechanics

59

As known, it is often possible to associate a classical density with a density operator p in the form Pc(j3) == (j3lpl,B) that must be positive and normalized (13 could be a coherent state). Therefore its differential entropy takes the form

Now in the case that the density operator can be expanded in the P(a)-representation (see § 1.2.8), another differential entropy can be introduced by using the usual form

S(P) = -

J

P(a)logP(a)J!a

Then the following inequalities hold (1.4.5c)

Finally, for Ai

S

> 0, Ei Ai == 1, it can be shown that

(2;:: AiPi) ::; 2;:: Ai S (Pi) - 2;:: Ai log Ai z

z

(1.4.5d)

z

After a measurement ak (that could be degenerate) with probabi~

lity Pk of an observable A ==

..-- (a)

Ek akIIk

of the system S by an

p,

a mixture of states of en-

apparatus I, the system evolves' from

tropy S(p) 2: 0 to {ik, (generally not a pure state) of entropy S({ik). The process should be examined on the basis of whether or not the measurement is read and divided into stages such that (1.4.6)

60

Introduction to Photon Communication

for which we have

{

sCPt) 2 Seji)

(1.4.7a, b)

S(P') 2 (S(Pi))

---(a).-. ---(a)

where (S(Pi)) == - EkPkTr (PilogPi) and Pi ==

Ilk

P Ilk

Pk

.

In fact, the measurement stage of the scheme (1.4.6), which is a deterministic one, yields no information. This is because it implies quantum correlations due to the interaction S - I which vanish as soon as the system and the apparatus are separated. During this stage, irreversible processes take place as a result of an energy transfer that affects the system under observation (e.g. quantum Zenon effect). Notice that although an apparatus is a collection of quantum microscopic elements, this macroscopic object must be deterministic for the outcomes to be reproducible. The reading (undeterministic process with no time evolution equation) which is a selection among all systems, informs us about

P'

instead of

p.

It

allows us to gain some information as seen in 1.4.7b.

1.5 References [1] [2] [3]

[4] [5]

A. Messiah: Mecanique Quantique (Dunod, Paris 1959). 1.1. Gol'dman and al: Problems in Quantum Mechanics, ed. by D. ter Haar (Infosearch Ltd, London 1960). R.J. Glauber: Quantum Optics arl,d Quantum Electronics, ed. by C. de Witt, A. Blandin and C. Cohen-Tannoudji (Gordon and Breach, New York 1964). W.H. Louisell: Radiation and Noise in Quantum Electronics (Mc Graw Hill Book Company, New York 1964). J.R. Klauder, E.C.G. Sudarshan: Fundamental of Quantum Optics (Benjamin, New York 1968).

I. Elements of Quantum Mechanics [6]

[7] [8] [9]

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

61

P. Carruthers, M. M. Nieto: Rev. of Mod. Phys., 40 411 (1968). T.W.B. Kibble: Quantum Optics, ed. by S.M. Kay and A. Maitland (Academic Press, New York 1970). R. Loudon: The quantum theory of light (Clarendon Press, Oxford 1973). S.R. de Groot: La transformation de Weyl et la fonction de Wigner: une forme alternative de la mecanique quantique (Presses de l'Universite de Montreal 1974). C.W. Helstrom: Quantum Detection and Estimation Theory (Academic Press, New York 1976) T.S. Santhanam, A.R. Tekumalla: Found. of Physics, 6583 (1976). H.P. Yuen: Phys. Rev. A, 13 2226 (1976). C. Cohen-Tannoudji, B. Diu, F. Laloe: Mecanique Quantique (Hermann, Paris 1977) part I. H.P Yuen, J.H. Shapiro: in Coherence and Quantum Optics IV, ed. by L. Mandel and E. Wolf (Plenum Press, New York 1978). C.M. Caves, K.S. Thorne, R.W. Drever, V.D. Sandberg and M. Zimmermann: Rev. Mod. Phys., 52 341 (1980). C.M. Caves: Phys. Rev. D, 23 1693 (1981). W. Thirring: Quantum mechanics of Atoms and Molecules. A course in Mathematical Physics 3, (Springer-Verlag, Wien 1981). W. Zurek: Phys. Rev. D, 24 1516 (1981). W. Zurek: Phys. Rev. D, 26 1862 (1982). C.W. Gardiner: Handbook of Stochastic Methods (Springer-Verlag, Berlin 1983). A.M. Perelomov: Generalized coherent states and their applications (Springer-Verlag, Berlin 1986). S. M. Barnett, D. T. Pegg: Jour. Phys. A: Math. Gen., 19 3849 (1986). K. Kraus: Phys. Rev. D, 35 3070 (1987). S.M. Barnett, P.L. Knight: Jour. of Modern Optics, 34841 (1987). G.S. Agarwal: Jour. of Modern Optics, 34 909 (1987). M. Hillery: Phys. Rev. A, 35 725 (1987). R. Loudon, P.L. Knight: Jour. of Modern Optics, 34 709 (1987). H. Maassen, J.B.M. Uffink: Phys. Rev. Lett., 60 1103 (1988). J. Rai, C.L. Mehta: Phys. Rev. A, 37 4497 (1988). H. Fearn, M.J. Collett: Jour. of Modern Optics, 35 553 (1988). C. Cohen-Tannoudji: Coherences quantiques et dissipation, Cours du College de France, Paris 1989-1990. S. Reynaud: Ann. Phys. Fr., 15 63 (1990).

62

Introduction to Photon Communication

[33]

T. Uyematsu, O. Hirota, K. Sakaniwa: Quantum Aspects of Optical Communications, ed. by C. Bendjaballah, O. Hirota and S. Reynaud (Springer Verlag, Berlin 1991). V.B. Braginsky, F. Ya. Khalili: Quantum measurement, (Cambridge Univ. Press, New York 1992). C.L. Mehta, A.K. Roy, G.M. Saxena: Phys. Rev. A, 46 1565 (1992). S. Abe, R. Ehrhardt: Phys. Rev. A, 48 986 (1993). L.P. Hughston, R. Jozsa, W.K. Wootters: Phys. Lett. A, 183 14 (1993).

[34] [35] [36] [37]

II. Performance Criteria: Detection, Information

This section deals with the criteria of the statistical detection and information theories. Basic results of both formulations, classical as well as quantum are recalled and demonstrated when it is necessary. Applications to cases of practical importance are treated in detail.

11.1 Classical ForDlulation Once the receiver operator is fixed, the system performances can be calculated using the results of the classical theory of detection and estimation for detection purposes. On the other hand, as the information criterion is closely related to entropy, before applying the theory to optical communications, it is useful to set out the main properties of the entropy function in terms of classical probability. We begin with the presentation of some elements of the theory of statistical detection.

11.1.1 Statistical Detection The standard theory shows that detection optimality is attained using the maximum likelihood ratio test. In this subsection, we propose to show some calculation of this test for photon counting processing in the case of PPM modulation format.

64

Introduction to Photon Communication

II.I.la Likelihood Ratio Test Let Y be the M -dimensional vector of the M photocounts {Ni },

1,

, M). The conditional probability distribution p (ylm),

m == 1,

, M denotes the probability that Y == y E lN M when

(i

the message m is sent. The test strategy separates the observation space INMinto M

+ 1 domains of decision defined by M

D m = {y E lNMlp(ylm) > p(yll) '11

i- m} and Dc = lN

M

-

U Dm m=l

(11.1.1) The test works as follows: - for y E D m , m is selected - for y E Dc, any message is selected with equal probability The probability of error Pecan be bounded by computing the probability that the observation falls in any of the M domains D m , so that'

(11.1.2) The hypotheses being independent, we have:

p(ylm)

= Pl(n m ) II po(ni) i=l i#m

Moreover, the domains can now be redefined using the likelihood ratio A(nj) =

:~~:~~. Hence D m

= {y E lNMIA(n m )

> A(ni)

Vi

i-

m }. Because A( n j) is a monotonic increasing function of n j, we set D m = {y E lNMln m

> ni

Vi

i- m}.

II. Performance Criteria

65

Therefore

p(y E Dmlm) = L

p(ylm) = L

L··· LP(Ylm)

yED m

(11.1.3)

Finally,

Pe

~ ~Pl(n) 00

(

1- [1 - ~PO(l)] 00

M

-

1)

(11.1.4)

This test consists of evaluating A( k) for the pa~ticular. measured photon number and then comparing it with the A, the threshold of the test.

II.l.lb Binary Hypothesis Testing In a binary detection problem, the decision consists of a choice between two hypotheses, Ho and HI (signal absent and signal present respectively) with the associated probability distributions Po and PI and a priori probability (0 and (1. In this case, the hy-

pothesis HI is chosen whenever the observed photon number in the time interval [0, T] : N(T)

= n, exceeds a

certain integer decision

level A, and the hypothesis H o is chosen when N(T) For N(T) = n

f(O

~

=

n

< A.

== A, HI is again selected with a probability

f < 1).

The threshold is chosen based on some criterion: if Qd, the probability of detection, is maximized subject to a constraint on the probability of a false alarm (Neyman-Pearson criterion), the probability of detection is given by

66

Introduction to Photon Communication 00

Qd = Pr{HtlHd = fpt(-\)

L

+

pt(k)

(11.1.5)

Po(k)

(11.1.6)

k=A+1

where 00

Qo

L

= Pr{HtIH o} = fpo(-\) +

k=A+1

is the probability of false alarm which is the probability of choosing "signal present" (hypothesis H 1 ) when "no signal" (hypothesis H o) is true. The required values of A and

f

are obtained as indicated

with a preassigned value of Qo and given po(k) and P1(k). The average probability of error is then given by Per

= ~(1 + Qo + Qd)·

As proved [6], for a Neyman-Pearson test the values of Qo and Qd completely specify the performance: the curves Qd versus Qo are called the Receiver Operating Characteristic "ROC". In the next section, we will briefly present the method of computing the ROC curves for the case of multiple hypotheses.

II.I.Ie Multiple Alternative Hypotheses Suppose the signal to be detected belongs to a set of cardinality

M, depending, for example, on a parameter

M-1 e == { () } 1=1 . We as-

sume that the detection process remains basically binary H o (signal absent) and H 1 one of the HI ; 1 == 1, ... , M - 1 (signal present). To simplify the presentation, we assume that all the hypotheses

HI are equally likely. It is convenient to define the test vector

4J(X) == (
def prob

[decide HdX

= x]

II. Performance Criteria

x E X so that 0 ~ cPI(X) ~

of the single observation X 1;

67

2:~o
With the Neyman-Pearson criterion used, we fix the probability of false alarm

Qo

=

L(1-


= x))po(x)dx

at some level a and seek the test function which maximizes the probability of correct detection

Here, the generalized likelihood ratio At(X)def :~i~~ is combined with a randomization to read

1,

max A,(X) >,,\ I

(11.1.7) 0,

A1(X) < ,,\

where"\ and II are chosen to achieve a preassigned value of Qo == a. As previously, for photon counting. purposes, the ROC curves are such that

Qd = Pr{HzIHz} = Izpz(A)

+

L

pz(k)

(11.1.8)

po(k)

(11.1.9)

k=A+l

Qo

= Pr{HzlHo} = fzpo(A) +

L k=A+l

M

o :os; Iz < 1 ; Lfz = 1=0

1

(11.1.10)

68

Introduction to Photon Communication

Before closing this brief introduction to the statistical detection theory, we wish to mention another criterion which requires maximizing the signal to noise ratio. In fact, let the random variable X be a sufficient statistic of the data and suppose we want to decide between two hypotheses that depend on the real signal parameter fl, HI : PI(x,fl), H o : Po(x,O). We may define the signal to noise ratio as

Ll = (E[XIH1 ] - E[XIHo])2 E[X 2IHo] - (E[XIHo])2 where E[XkIH i ] imises L1 for fl

-t

=

J Xkpi(X)dx, i = 0,1.

(11.1.12)

A processor that max-

0 is called a threshold detector.

11.1.2 Information For a discrete memoryless channel without cost constraint which is the conventional modelisation of an optical channel, a few fundamental results are required to derive the expressions of some information criteria: capacity, cut-off rate and rate distortion function. We begin with briefly recalling some elements of the classical theory and then extend some results to the quantum case in the next section.

II.l.2a Entropy and Algorithmic Complexity The axiomatic approach, rather mathematical, to justify the concept of entropy and derive its properties, will not be mentioned here. The statistical thermodynamics point of view is first adopted. The probability Pm for a system to be in state 1m) characterized by the energy Em, yields the entropy S

= -k B

L mEM

Pm logPm, with

II. Performance Criteria

Em Pm

= 1,0 :::; Pm

<

1,Pm =

p:n,

69

and where kB is the Boltz-

mann's constant. To treat an example, let be given an: ensemble of N noninteracting particles (e.g. spins {O'i} such that (O'i) = 0'). Let us consider that q+ = Nt n particles (resp. q_ = N-;n) are oriented up (+1) (resp. down (-1)). The total number of the possible sequences of such order obeys the Bernouilli law Wen) = (::)( ~ )q+( ~ )q-, (q+ + q_ = 1). Under the action of external field of intensity 1, each particle (i, O'i), has a probability Pi( +1) (resp. Pi( -1)) to be oriented so that the total energy U = E~l CTi = N (pi ( +1) - Pie -1)), from which we deduce

Pi(±l) =

t ± 2~·

To calculate W

=

p(b) ,

it is useful to utilize the partition func-

f3u = II:' '" f3ui = '" = (e- f3 + e(3 )N = LJ ui =±l ea=l WUi=±l eEN=o (~)e-(3(2q-N), where f3 = Ie 1 T' T is the temperature. Assum-

tion Z

ilU are of equal probability, we may

in; that all states of energie U ± deduce that W rv Z rv(~) with q± Using the Stirling formula, log W

rv

U~N = Npi(±).

= log N! -log q! -log(N - q)! = N log N - q+ log q+ - q_Iog q_ = N (log N - Pi( +1) 10g(Npi( +1)) - Pi( -1) 10g(Npi( -1))) = -N L p(CTi)logp(CTi) = ~S Ui=±

which is of the same form as the previous definition.

It should be stressed that despite the fact that there is a close correspondence between thermodynamical and statistical quantities, their relationship with the information entropy is not clearly established. We now introduce the concept of entropy as a measure of uncertainty of a random variable X : the random variable q( x) = log

S(X) =

L xEX

(x, p( x))

as expected value of

plx)

p(x)q(x) = -

L xEX

p(x)logp(x)

70

Introduction to Photon Communication

With the aid of the joint probability p(x, y) and the conditional probability p(ylx) of a pair of random variables X, Y, the above entropy is readily extended to the joint entropy S(X,Y) = - LP(x,y)logp(x,y) x,y

=-

and to the conditional entropy S(YIX)

Ex,y

p(x, y) logp(ylx).

It is therefore simple to introduce the mutual information

I(X; Y)

= S(Y) -

S(YIX)

which is a quantitative measure of information one random variable contains .about the other on one hand, and the relative entropy that characterizes the distance of two probability distributions S(pllq) ==

Ex p( x) log :~~~

on the other hand.

Another extension concerns a continuous random variable for which the differential entropy can be defined as Sf(X) = - L!(x)log!(x)dx, L!(x)dx = 1

This function has, however,·a limited interest because it can diverge or can be negative. Nevertheless, it is often justified to simplify the calculations as shown in the next conventional example. For a one dimension time-dependent system the following constraints

Ix xkp(x, t)dx

{X : x,p(x, t)}

for which

== mk, k == 0,1,2, are given

such that ma == 1, ml (t), m2, it is readily shown that the maximization of Sp(t) == -

Ix p(x, t) logp(x, t)dx,

the differential entropy, us-

ing the method of Lagrange multipliers yields the Gaussian distribution

p(x, t) == N(ml, u) where the variance is u 2 (t) ==

m2 -

mi(t).

II. Performance Criteria

71

To do so, we may interpret ml(t) = ml(O)e-,Xt as the solution of

the first order differential equation :t ml (t) · Itt equlva en 0 Aml

2

-

8p(x,t) dml 8mI dt

+ 'xm 12 8 28xp(x,t) 2

8p(x,t) 8x

::2 (>.m

8p(x,t) 8t

+ >.ml (t) =

_ _ (8 P(X,t) 8x

and leads to

8p(x,t) 8t

0 which is

+ ml 8 2p8x(X,t)) 2 = .2.... (Ax p(x 8x

dmi dt

'

t))

+

p(x, t)), the well-known Fokker-Planck equation (FPE). To

prove the result, it is useful to notice that, for a normal distribu-

tion p(x,t)

= N(ml,u),

p(x,t)

= -(x -

mdap~:,t) - u2a2;~~It), and

8p(x,t) _ _ 8p(x,t) _ m 8 2 p(x,t) 8mI 8x 1 8x 2 •

Since the difficulties due to the definition of differential entropy disappear when studying relative entropy or mutual information, it is therefore, preferable to deal with relative entropy and not to define differential entropy at all. In that case, a general rigorous theory can be developed from the point of view of the measure theory [29]. As it is just seen, the definition of entropy requires the knowledge of the probability distribution of the observable. However, the concept of probability, that is essential in quantum mechanics, is not necessary in classical physics which is a deterministic approach. That the reason why, certain authors (e.g. Zurek [27]) reinterpreted the usual concept of entropy as the algorithmic complexity to specify the state of the system under observation, without utilizing any notion of probability. Let us briefly introduce the concept of algorithmic complexity in order to see the sort of link it is made with the entropy. Let

C(p) be the output of the computer C processing the numerical program p. Now, let l(p) denote the length (integer) of a finite

72

Introduction to Photon Communication

length string X

def

x. The algorithmic complexity that depends on

=

C via a constant Kc(X)

min l(p) is the minimal length of a

C(p)=X

program to obtain X. As a first approximation, the complexity can be taken equal to the length of the string. Example: Let us order a sequence of N positive numbers a(k) ~ 0, k = 1, ... ,N, each number requiring, e.g. log2 am to be coded. The output string {a( k)} is ordered so that the numbers are in increasing values. Therefore [«X) rlog2 Nl + Nrl og2 am 1 (rxl is the smallest integer ~ x). f"V

Remarks: (i) From the Kraft inequality of the log function in base 2),

Ei 2- 1i

~ 1, and the convexity property

Ei Pi (2;;;) ~ Ei Pi log 2;;;), (log taken ~ 1, log Ei Pi (2;;;) ~ 0, then

(log

Ei Pi (2;;;)

2- 1i

~pdog

p:- ~ 0

---+

E[li]

~ S(Pi)

t

(ii) The probability of a string x [32, p. 160] can then be introduced:

Pr(C(p) = x) =

Ep:c(p)=x

2- 1(p).

Tp.e relationship between complexity and entropy is established for i.i.d. random variables {Xi}:1 : Xi,p(Xi)

for which we denote

X : x,p(x) = n~ p(Xi). The theorem [32, p.154] states that

lim E [K(XIN)]

N-+oo

where S(X)

=

= S(X)

N S(Xi) is the entropy of the stochastic pro-

cess X. For a given l(X)

=

x, the complexity is written as a

conditional complexity Kc(Xlx) from which it will be derived

I(X; Y)

= K(X) - K(YIX)

that is interpreted as a quantitative

measure of information about Y contained in X.

II. Performance Criteria

Notice, however the equality I(X;Y)

=

73

I(Y;X) cannot be ex-

pected to hold exactly. A bibliography concerning the questions of entropy and complexity and their connections and relationships with other fields can be found in [32, and references therein]. Notice, however, that it seems difficult to formulate the notion of algorithmic complexity for quantum systems [30].

II.l.2b Channel Capacity Let the discrete channel input sequence be X

=

{Xi

given with the probability q(Xi) and the output sequence Y

= i}~l = {Yi =

j}.f==l. The statistical description of the channel is determined by the II matrix N x M of elements being the conditional probability distribution p(Yilxi). The mutual information between X and Y can be written in the usual form

I(X, Y)

def

S(X) - S(XIY) M

=

N

L L q(xi)P(Yilxi) log L: i=l j=l

(

I) ) (,1

(11.1.13)

P Yj Xi k

(

q

Xk

P YJ

Xk

)

The channel capacity is the maximum under the constraints Q of the mutual information rate (11.1.13) C = max I(X; Y) Q

M

(11.l.14a, b)

Q: Lq(Xi) = 1 i=l

Theorems on convexity of the mutual information, are demonstrated in various textbooks (see [3,29]). Recall that I(X; Y) is convex n with respect to q(Xi), 'Vi and convex U with respect to p(YjIXi), 'Vi,j. Let us show here a simplified calculation [8]. We use a Lagrange method for the maximisation of (II.1.14a,b) to seek the probabilities {q( Xi)} such that

74

Introduction to Photon Communication M

Vi,

8~~i) = 0 where A = S(X)-S(XIY)+A?:q(Xi) -1. After simple 1=1

algebra, it is shown that the system (II.l.14a,b) reduces to

(II.l.14c)

where C~f 1 - A is called the channel capacity. C is exactly the channel capacity because for q( Xi) satisfying (II.l.14c) we do have that

L:i q(xi)I(xi; Y) = C ~f maxI(X; Y). To determine {q(Xi)}, suppose that M = N and assume that the matrix P of the transition probabilities p(Yj IXi) is not singular. Therefore M

Vi = 1, ... ,M

L

[C

+ logp(Yi )]p(Yil xi ) =

-S(Ylxi)

j=l

M

p(W) = Vj

L q(Xi )p(W IXi) i=l

= 1, ... ,M

M

LP(W)

=1

j=l

(II.l.14d, e) Inverting (II.l.14d,e) yields

C = log

~ exp ( -

q(Xi) = exp( -C)

t

1rii S(YI Xi))

~ 1rii exp ( -

P(Yi) = exp(-C)exp (-

t

t

1rik S(YI Xk))

(II.l.15a, b, c)

1rii S(YI Xi))

where 1rij are elements of P- 1 . However, it may happen that q(Xi) < o. To avoid such case and guarantee that q(Xi) ~ 0, the standard KuhnTucker conditions [32, p.462] can be used to characterize the maximum. They are equivalent to aqtxi) Alq(Xi) = 0 for q(Xi) > 0 and aqtxi) Alq(Xi) ~ o for q( Xi) = 0, where A is the Lagrangian function.

II. Performance Criteria

75

Here M

Q': Lq(Xi) -1 ~ 0,

-q(xi) ~ 0

(II.1.15d)

i=l

must be substituted into (II.1.14b) and numerical calculation are very simple. The algorithm of computation is summarized as follows. q( x) = 1 and Let us be given the vector q = q( x) : x = {Xi} EX, the matrix Q = Q(xly) : y = {Yi} E Y, Q(xIY) 2: 0, Q(xIY) = 1 \ly.

2:x 2:x

2:x,y p(ylx)q(x) log QJ(;)') , we first get a~~(~~) = Ai = L: Yi p(Wlxi)logQ(xilw) - L: Yi p(Yilxi)(logq(xi) + 1) = L: Yi p(W IXi) log Q(xiIYi) - (logq(xi) + 1), Ai being derived from L:x q( x) = 1 to yield q( Xi) given by R2 and then Define F(q,Q) =

X'.I exp

q(k+1)(Xi)

(LP(Yilxi)logQ(k)(xilw))

(Rl, R2)

= ~~~Y~j~~~~~~~~~~

L exp (LP(YilxD IOgQ(k)(x~lw)) x~

Yj

Starting from arbitrary

q(O)

(e.g. 1/M), the steps of the computation

follow ( ) qO(x)

(Rl)

)Q

(0)

(xly)

(R2)

)q

(1)

(1)

(x)~C(l)=F(q ,Q

(0)

)

so that C(k+1) = F(q(k+l),Q(k»). For € such that IC(k+l)-C(k)1 the convergence can be proved lim IC - C(k)1 = o.

< €,

k~oo

For several types of channel, efficient numerical algorithms are available and can be easily implemented [32, p. 364]. As for the above discrete case, a simplified version of the capacity calculation can be presented [32, p. 241]. Let us assume that a channel superimposes an independent white Gaussian noise Z, g( z) E N(O, lT~) to a continuous source X of a probability density f(x),x E X and a differential entropy SJ(X).

76

Introduction to Photon Communication

The channel output is Y

hey)

= f(x)g(z),

(y2)

= X + Z and we have

= oJ + (X 2) = No + P,

= ~ log27re(T~

Sg(Z)

(11.1.16)

The channel capacity

C

f(x)

clef

= max (C)

I(X; Y)

where (C)

=

L

~ 0,

L

f(x)dx

2

x f(x)dx

~P

We now use the fundamental lemma which yields Sf(Y) given P, No, to obtain

I(X; Y)

= S(Y) -

S(YIX) = S(Y) - S( ZIX) 1 log 27re(P ~ 2"

= S(Y) = S(Y) -

=1

(11.1.17 a, b) for

:s; Sg(Y),

S(X + ZIX) S( Z)

P) + No) - 2"1 log 27reNo = 2"1 log ( 1 + No

which finally yields

c = ~ log ( 1 + :0)

(11.l.18a)

The capacity is attained for X E N(O, P). When the constraint on the

bandwidth 2W of a white noise is included,

c = W log (1 + N~W ) W

--+ 00,

W

--+

C

O,C

--+

"-J

:0

Wlog N~W

= Wlog

:0 -

WlogW

(11.l.18b)

--+

0

For a colored Gaussian noise [7], (11.l.18a) becomes

C

where P

=

I:

=!

2

1 (1 + 00

log

-00

IS (v)dv and N

=

i:

IS

(V)) dv

IE (v)

(11.l.18c)

IB (v)dv.

To continue with the example, let us specify the type of the input source X as M orthogonal signals. The decision rule which is optimum in the sense that the probability of error is mimimum defines the domains of decision {Am}.

II. Performance Criteria

77

For equal a priori probabilities

Am

={ y: ={ L

logp(ylx m)

> logp(ylxm/) }

2(xmn - xm1n)Yn > L(X~n -

n

x~/n) }

n

for 'Vm' =I m. To calculate the probability of error, we first study the case M = 2. We notice that Pc = p(y E AlIHl ) = f fA! f(Yl, Y2 )dYl dY2. Thus

Pc

1

= 211"N,

o

JOO

dYl exp( -

(Yl - VE)2 2M ) 0

-00

jYl dY2 exp( - 2N,Y~ ) 0

-00

= - 1 JOO dOl exp( -Or- )1- j~+(h d0 2 exp( - -(J~) Vfi

2

-00

= - 1 JOO exp( Vfi

-00

Vfi

2

0 ) ( 1 - Q( 2

2

-00

II ) + 0)

-

No

dO

For arbitrary M, using the M -ary property: the signal is transmitted only if Xi ---* Yi, such that given Xi, we have Yi > Yj, 'Vj '# i and

so that

Thus M - 1 "equal" components contribute to the total probability

Noticing that for
Pe = 1 - Pc

= 1-

JJOO

< (M - 1 ) - -1 .

-

OO

-00

p(YIHl)(

Vfi V. INOI

e -~d(J 2

jY

M-l

-00

p(xIHo)dx)

< M-

-

1 e _JL 2No

fi[; Vu.NQV ~1f

dy

~

=

78

Introduction to Photon Communication

To study the asymptotic case M ~ 00, we set the following notations M = eRT,R = ~logM,E = ~logM with E ~ PT,

P e (M)

£(M)

= M1-i

R { R

>C

rv

:C R C

1- e-cE(M)

<1

£( M)

rv

M 1

< C: R > 1 £( M) = M

~ 00 ---+

0

Pe

~

0

(11.1.19)

Before closing the section on capacity criterion, let us just express the fundamental theorem channel coding: Given M

= eN R

~

00

equally likely signals, a constant C exists,

called the Gaussian channel capacity whose properties are such that

- R > C, the P e probability of error

~

1

- R < C, the P e probability of error achieved with optimum receivers

~

0

Rigorous demonstration of the capacity theorems (direct and converse) for Gaussian channel can be found in [3,5]. Remark: Notice that the shape of the signals is not important. We here again mention that the above discussion is only a first approach.

II.l.2c Cut-Off Rate To deal with the cut-off rate, we first define some notations and simplify the presentation of some results. For each pair

Xl, x2

taken from a finite set, the source alphabet Ax,

supposed to be i.i.d. with common distribution function Prob(X ==

x) == Q( x) called the source statistics, we introduce

J(X},X2)

=

L yEAy

y'p(ylxdp(ylx2)

II. Performance Criteria

79

where the finite set Ay is called the destination alphabet. Let

Jo = min {E[J(XI' X2)]} Q(x)

Consider a code of length N containing two codewords Xl =

{Xli}~l' X 2 = {X2i}~I· Assume that given a received N -uple Y, the decoder output is

- Xl if p(YIX I ) > p(YIX2 ) - X 2 if p(YIX2 )

> p(YIXI ).

Let YI = {Y : p(YIXI ) > p(YIX2 )} and Y2 = {Y : p(YIX2 ) >

p(YIX1 )}. If

Pe(i)

denotes the probability of error of decoding,

given that Xi is sent, we can show that

P e (i) ~

L p(yIXi) ~ L yE~

N

vp(YIXI)p(YIX2 )

=

YEA~

II

J(Xlkl X2k)

k=l

(11.1.20) N

The generalisation to M codewords is P e (i) ~

L IT J ( j=l j¢i

X ik 1 X

j k ).

k=l

Furthermore by averaging over all codes, we obtain E [P e (i) ]

<

M exp( -NR o). Finally, the cut-off rate in nats units, is given by Ro = - log ( J°)

(11.1.21 )

With regard to this cut-off rate function, the important result proved by McEliece [18] can be summarized as follows For a DMG and for any rate less than the cut-off rate R < Ro, a code denoted {Xl,X2, ... ,XM} exists with at least M = exp(RN)

80

Introduction to Photon Communication

codewords of length N and an appropriate decoding rule, such that if

Pe

= it E~l P e (i)

denotes the average decoding error probability,

then

P e < exp(-N(Ro - R))

(11.1.22)

This theorem: - is stronger than the channel coding in that it gives an explicit estimate of how small Pecan be made as a function ofn.

- is weaker in that VR, R

< C.

Therefore for rates R o < R < C it does not imply that P e

---+

0 is

at all possible.

II.l.2d Rate Distortion The purpose of this section is to briefly summarize some elements of the theory of the rate distortion functions for discrete memoryless sources. We begin with introducing notations and simplifying the presentation of a few results. Consider a discrete, noiseless, memoryless channel that maps the source (X

= {j}tt,

Prob (X

(X =

{k}f, Prob

(X =

q(klj),

Vj, Ef=l q(klj) = 1.

k)

=

= j) =

pj)

into the destination

r k ) by the transition probability

Suppose that for each pair (j, k), a nonnegative number L1 j k exists which measures the degradation (error distortion) that the symbol j is reproduced by the symbol k and does not depend on any other

terms. Its average is

(Ll jk ) =

L LljkPjq(klj) = d(q) j,k

(II.1.23a)

II. Performance Criteria

81

The mutual information is as usual

I(X;X)

= I(q) = LPjq(klj)log-q(_kIJ_O) j,k

(11.1.23b)

rk

Now, given Pj, djk and D, the R(D) rate-distortion function is simply

R(D) { QD

= qCQD min I(q)

= q(klj) : d(q)

(11.1.24) ~

D

the minimum does exist because the function I( q) is convex U with respect to the q(klj). Thus, the new source X will reproduce some statistical properties of the initial source X with lower entropy. The

R(D) is the minimum bit rate required to encode X subject that the distorsion is maintained arbitrary close the level D. Assuming that M

= N and using the well-known method of the

Lagrange multipliers and the conditions for making sure that all probabilities are positive and normalised [32, p. 462], the appropriate solution is determined such that

s
Ck = E j Ajpj exp(sLljk) ~ 1, Vk q( klj) = Ajrk exp(sLljk)

D

(11.1.25a)

= Ej,k djkPjq(klj)

Dm

= LPj min (Ll jk ) , j

DM

= min ~PjLljk J

where D m and DM are the minimum and the maximum values of D respectively. A more general expression of R(D) involves real matrices formulation.

82

Introduction to Photon Communication

= pjq(klj)} obtained from (II.1.25a), A the matrix of elements {ajk = e sL1jk }, {Wjk} elements

Denote II the matrix of elements {7r"jk

P the vector of elements {p j }, Ll the

of the inverse matrix A -1,

matrix of elements {Ll j k }, then

R(D) = S(p)

+ sD(s) + LPj logJLj

(II.1.25b)

j

where S(p)

,\"P" J J

EkWjk.

For the distorsion measure selected here Lljk

=

Lllj-kl

which includes the probability of error criterion Lljk

=

=

1-

Ll 1 bjk,

Eq. (II.1.25b) can interestingly lower bounded. To find the bound, we use the fact that for some we have

ajk

=

al

such that

ao

=

1

2::

lsi> Is*1

a1 ~ ... ~ aM.

~

1,

The cal-

culation of the inverse matrix A -1 becomes easy yielding {Il j}

rv

{ (1- (M -1)a 1)2} Vj, (M ~ 1). The average value of the distorsion will then be given by D *() S

Ilj sL1"k = "L-i WkIPl-~jke A

J," k , I

J

III

(II.1.25c)

"

rv

(M - 1)a1 L-iWkIPI

rv

(M -1)a1 1 + (M -1)a1

k,l

where

a1

=

eS •

It leads to a parametric expression of the rate

distorsion

R(D*)

rv

S(p) + sD*(s) -log (1 + (M -1)a1)

(II.1.25d)

II. Performance Criteria

where s(D) = log (M-lf
83

Is*I'" S(DM).

We finally summarize a few basic properties of R(D) that will be utilized in the present context, among those proved by Berger [11] - There is one and only one relative min of the mutual information with respect to the transition probability.

- R(D) is a continuous, monotonic decreasing, convex U function in the range D E [D m , DM].

- R(O) = S(p), R(D) = 0 for D ~ DM (p, the source a priori vector probability). We close this section by treating two examples for Lljk and where the previous approximate calculations become exact. First, for M = N = 2, Ll jk = 1 - 8jk , given Po = P,Pt = 1 - P, it is proved that ro

=

t~2~' rl

= 1-

ro, (0 ~ ro ~ p ~

t). It is then

easily derived that

R(D) { R(D) where S(z)

= S(p) =0

S(D)

o ~D

(II.1.26a)

= -zlogz-(l-z)log(l-z). Secondly, a generalisation

to higher number of messages will show that the output vector probability

--:t

can be written formally as a linear function of the

input vector probability

p

--:t = 1 + (M 1-

at

l)Ul

P_

1

Ul

1-

(II.1.26b)

at

which yields

R(D) = S(p) - Dlog(M -1) - H(D) for 0 ~ D ~ DM { R(D) = 0 for D ~ D M (II.1.26c)

84

Introduction to Photon Communication

where DM

= inf[1- m~xPj, (M -1)m~npj] ~ 1- M . 1

J

J

We refer the interested reader to the books [11,18] for more details.

11.2 QuantuIn Expressions As has been done above for the classical formulation, the quantum expressions for the detection and informational entropy criteria will be briefly established. Let us consider the system of communication [4,10] in which the

input source of messages, the density operators {¥il~Ol emitted with the a priori probability and the measurement operators verify

Vi,

¥i 2: 0, Tr (¥i) =

1

M-l

s=

ei 2: 0, L ei = 1 i=O

(II.2.1a)

N-l

Vj, Rj

L Rj = n

> 0,

j=O

The detection channel is characterized by the transition probability

Pij

= Tr

[¥iRj], Pij 2: 0, LPij = ei, LPij = 1, N j

=M

ij

(II.2.1b) Hence the probability of error will be M-l

P e = 1- L j=O

ei Tr

[¥iRj]

(II.2.1c)

II. Performance Criteria

85

The problem of quantum optimal detection as formulated by Helstrom and Holevo is to find the operators

{Rj}

so that the P e is

minimum. Although considerable work and important results have been obtained over the last twenty years, a rigorous solution to this problem has not yet been given. However, for some interesting cases, exact results have been established. They are the discrimination between

- M linearly independent pure states - M commuting density operators. The detection problem is to make a choice, after a single measurement, between M hypotheses Hi each corresponding to C be the specified Bayes cost matrix. The element of choosing hypothesis Hi when Hj is true Vi has to minimize the average C

i= j,

Let

is the cost

Cij Cij

{¥d.

> C jj . One

= L:ij Cijei Tr [¥Jlj].

The

{Rd

making C* == C min is called optimum detection operator (o.d.o.). Let us denote

Y = L:: j Rj¥j, Wi = L:: j ejCij¥j and h = Wi - Y.

The necessary and sufficient conditions to verify, Vi,j, are

~ = l:j R~¥~, C~ ~ {

ri 2:: 0,

Tr[Y]

(II.2.2a, b)

riRi == Riri == 0

11.2.1 Decision Among M Linearly Independent Pure States We set

¥i = I'l/Ji) ('l/Ji I·

Taking the costs

Cij = -8i j,

timum p.o.m. is the set of projection-valued operators

Irj)(rjl, L:~1 ii j

= n, (rklrj)

the op-

iij

=

= 8kj, given the matrix rJt of the

scalar products of the input states (¢j I¢k) == I ==

e-f, Vj

i= k.

86

Introduction to Photon Communication

Demonstration that projectors are solutions of the system of equations can be outlined as follows [16, p. 117]. (i)

from II.2.2a and for IFj).lI'l/Ji)

(ii)

from II.2.2b

". . . . (~".......) Ri Y - ei1/li IFk)

=0

((1PiIFk)

= <5ik) . ,)

YIFj) ex Rjl'l/Jj) ". . . . ". . . . RiRkl1Pk)

(iii)

IWk) = Rkl1/Jk) ~ Rilwk) = 8ikl wk)

(iv)

With Ill}

= Ek uklwk}, and from

". . . .

= 8ikRil1/Jk)

(ii) ~ kRilll}

= 8iil ll }.

As a simple application, we consider an M -ary channel for which we set (wil1Pi) = a, Vi and (Wjl1Pk) == b, Vi ~ k. We have (1Pjl1Pj) == 1 = Ei(1Pjlwi)(Wil1Pj) == a 2

+ (M -

l)b 2 and also (1Pjl1Pk) == 1 =

Ei(1Pjlwi)(Wil1Pk) == 2ab + (M - 2)b2 • The probability of error can be written as Pe = where Xii

it E~l (I-Ix

= (Wil1/Ji) are elements ofthe matrix X

2

kk 1 )

for which xt X

=

tIt. We easily derive Pe =

For M

M-l 2

M2 b

--+ 00,

( = M-l M VI + (M - Ih -

~)2

V 1 - '"Y

(II.2.2c)

we again use the method of section II.l.2b where R ==

lo~M is the rate of information. We prove that for high intensity

11 2

>> R, b2

2

rv

T' by expansion of the first

asymptotic form of Pe will be given by

square root, that the

t M l - 2u , u = ~.

The information performances of this orthogonalized-signal coding channel will be calculated below. For M == 2, the probability of error is then given by (II.2.2d)

II. Performance Criteria

where 'r

= (~Ol'l/Jl)

87

which can easily be derived for 'r E R with the

help of Fig. 11.1.

Iro}.. · ·

.~~/~/

Fig. II. l.For M=2, a simple geometric representa-

~~~I'l/Jo)

tion is helpful. We consider

---t------..-~~~ .J.p.~)

.

that all states are normed to 1, lying on a plane P. We limit ourselves to given

o ~ 0 ~ t. p

In the plane P defined by the given input states (l'l/Jo),I'l/Jl); 8), the states 14>0),

IFo) and 14>1),IF1) can be determined such that 14>0)

1.

. 14>1), and I~o) .1 IF1). In fact, it is easily seen that Pe

2 2 = min ~(aio + a51) = min -2 (cos (8 + 8) + sin 8) 6 2 6

1

the minimum is satisfyed for ito that P e

=

= Iro)(rol

!(1 - sin 6) with cos 6 =

and 8

= !(~ - 8), so

'r).

Notice that Kennedy [13], Shapiro [22] proposed an interesting system that can be near-optimum. Basically, that receiver is so that

ito

= l"po}("pol that leads to the channel of Z-type.

For a non orthogonal resolution of identity e.g.

ito

= l'Pl}('P11),

('P11
=

E,

such that 0 ~

E

(ito

= I
~ ~), we show that

88

Introduction to Photon Communication

the probability of error becomes P~

= cos2 ( ft 8 )

which is larger

than Pee For M

== 3, recent calculations [35] suggest a practically feasible

system can attain a near-optimum probability of error. That proposed detection scheme is characterized by the set of projections

iil = IWl)(Wll,

ii2(fJ) = IW2(fJ))(W2(fJ)\, fi 3(fJ) = (w3(fJ)I(W3(fJ)1 = n- fil -

(II.2.3a)

fi 2(fJ),

with (II.2.3c)

In the above detection scheme, the parameter () is optimized so that the average error probability is minimized. With a geometrical interpretation, it is easy to see that this scheme specifies a measurement state IWt) along the state vector lat), and chooses the other measurement states IW2(f))) and IW3(f))) on the plane (P) which is perpendicular to the vector lat) at its origin. Therefore () is the angle (la2)p, IW2((}))), where ta2)p is a projection of la2)

onto the plane (P). Hence, we have: Pr(211) = Tr(¥lii2 (fJ)) = 0 and Pr(311)

= Tr(¥lii3 (fJ)) = o.

II. Performance Criteria

89

Furthermore,

Pr(III) = Tr(~ljjt) = 1, Pr(212)

"'" 1I"'" 2(B)) = [1 -1/121 2] = Tr(y,2

Pr(313)

= Tr(~3jj3(O)) = Id* sinO -

cos2B,

r'

vI-I/131 2 -ldl 2 cos O

(11.2.3d)

where

* d = /23 - /12/13 -1/121 2

(11.2.3e)

VI

Let us assume that the prior probabilities are equal, i.e.

ei = 1/3 Vi,

then the average error probability is given by

P e ( 0)

= 1 - ~ [ Pre 111) + Pre 212) + Pre313)] =

~ [1/12 12+ 1/131 2+ Idl 2 + 21 sin20 + 2g cos 0 sin 0]

=

~ [1/12 12+ 1/131 2+ Id/ 2 + 1 - Vj2 + g2 cos(20 +
where

2 +11131 2) I=I-Idl 2 -21 (1/12/ 9

= ~e( d)VI -

1/131 2

-

Idl 2

Then the optimum value B*, which minimizes (11.2.3f), is easily found such that the minimum average probability of error is given by -

-

1

[2

2

2

Pe = Pe(O) = 3 1/121 + 1/131 + Idl -

1

g2

+ Vj2 + g2

]

(II.2.3g)

90

Introduction to Photon Communication

Especially, if

lij ~

0 Vi

j, we have ()* ~ 0 and d ~

=1=

123.

The

average error probability can then be approximated by

(11.2.3h)

This is only twice as large as the one obtained by the average detection scheme. In this sense, the proposed detection scheme behaves asymptotically like the average error probability processing, and can be calculated very easily. Notice that because it is often difficult to evaluate the performance of the optimum processor, it is interesting to seek approximate solutions as close as possible to the optimum one. We just discussed "near"-optimality approaches. Now and in analogy to the clp,ssical theory, the quantum threshold detector can be considered. Define for i == 0,1

E[X IHi ] { where

it

Pl (J-l)

rv

(11.2.4a)

is the Hermitian operator maximizing Ll (11.1.12).

To obtain the equation for

o in

= Tr (pJi) Po + IJL Pl (J-l) IJL-+O

it,

we first assume that Tr

(poll)

the denominator of the expression giving Ll. Then for any

Hermitian operator that .d( ll)

(il t = il) and any real number ..\, we require

= .d( it + ..\il) which yield Tr

(2(Pl - po)il- (Poll + llpo)il)

= 0

so that (11.2.4b)

II. Performance Criteria

91

It can be seen that the solution is of the form (II.2.4c) It should be emphasized that this operator is only suboptimal compared with the o.d.o. The equation (11.2.4b) can also be obtained by using the symmetrical logarithmic derivative of Pt(p), given by (II.2.4d) Helstrom [16, chapter VI] treated in detail the case of detection of coherent state in a thermal state and gave an interesting physical interpretation of the quantum threshold operator. Using the expressions of

Po

and

Pt

(see 11.3.5), it is established

tl£P1(J.l)I/l-+o = (~) (apo + poat) ""'''''' apo

== e -w""'''''' poa =

P'oat e-watpo

(II.2.4e)

it = (N}+t (a + at) It is important to notice that this quantum threshold operator is proportional to the position operator

at

defined in (1.3.1) for the

coherent detection. However, this is valid only for p

~

O. For higher

values of p, the formula (II.2.4c) is more appropriate but the interpretation in terms of the criterion (11.1.12) is no longer valid. Detailed analysis and application of this operator to various cases have been presented by Yoshitani [9]. Another application that is the quantum phase state, is given below.

92

Introduction to Photon Communication

11.2.2 Application to the Quantum Phase State As has been done for the detection criterion, some elements of the quantum estimation theory are now recalled bearing in mind an application to estimate a quantum phase [16]. Suppose we want to estimate a phase

<jJ

E] -1r, 1r[. To treat this ptoblem,let the input

field defined by {ao,I7/;)o} and $ be some shift operator. Let the shifted field given by {a, 17/;)} and dii(
17/;)

=exp(i$aJao) 17/;)0

p(I$)d = (7/;ldii(
dii(
t

= I;=-7r dii(
and we seek dii(
=

L:~=o exp(in<jJ) In), we obtain

dii( ), I7/;) ) (exp( i
(II.2.5b)

To prove this result, some basic equations are briefly recalled. Let () E

e be the parameter to be estimated. Its probability density

a priori is given by z( 8), and let () be the estimate. The p.o.m is

denoted by dii(if). The cost is C(if,9)

where

W(8)

=

= -8(8 -

9) whose mean is

Ie p( 9)z(9)C(0, 9) is the risk operator.

Using the Lagrange multipliers method, we set the equations of the optimization with z( 8) ==

1 2 1r

II. Performance Criteria

93

[f - W(6)] dii(O) = 0 f - W(6) f defined so that

=

ft = f.

~0

(II.2.5c)

Ie W(8)dii(6)

The p.o.m.

dii( cP)

thus derived will be

called optimal under the maximum likelihood ratio criterion. Here we obtain

Now under a unitary transformation, the density operator of the

system will be given by p(O)

=

exp( -iNO)po exp(iNO), we have

from (II.2.5c)

[f - p(8)]~(8)

f -

= 0

p(8) ~ 0

(II.2.5d)

r = J~1r e(if)p(if)dii defined so that

~(8) = exp( -iN8)~o exp( iNii)

{

(II.2.5e) ~o =

211r1,)(,1

where Nlk) = klk). The state

-

I,) is defined by its coordinates {,d

in the space spanned by {Ik)}, with

94

Introduction to Photon Communication

Finally the posterior probability density function of the estimate

1f

of the displacement () is given by (11.2.5f) where q(
11.2.3 Communication with M Coherent States We present exact results for the noiseless channel and some approximate calculations for channels with thermal noise of weak power.

II.2.3a Noiseless Channels By analogy to the classical channel, it is possible to define the mutual information in the form

I(S,pj;Rk) = S(L~jq(klj)) - L~jS(q(klj)) j

(11.3.1a)

j

where E == {ej}~-l is the vector of input a priori probability and

II. Performance Criteria

95

(111.3.1b) is the transition probability: Vj, q(klj) 2:: 0, I:kq(klj) == ~j,

I:k q(klj) == 1. Because of the nonlinear form of the entropy function S, it is, in general, very difficult to determine the output measurement opera-

tor Rk that maximizes (II.3.1a) for given input states Pj. However, an important result demonstrated by Holevo [12,14] states that For given

ej

and {Pj}, the mutual information given by (II.2.4 a)

is upperbounded so that M-l

-L

~jS(Pj)

(11.3.2)

j=O

the equality being reached for

Pi

pairwise commuting.

The demonstration is based on Jensen's inequality generalized to operators. Extension of this theorem to a countable number of input symbolsM

--+ 00,

and a separable Hilbert space (d

--+

(0) has recently been proposed by Hall and O'Rourke [34]. This approach closely follows the classical theory but using an undefined measurement operator, can be considered as semi-classical. The same approach is developed in the next section for the study of the cut off rate.

II.2.3b Cut-Off Rate Criterion An information-theoretic criterion, that is, the cut-off rate, has interesting properties, as already pointed out (see § II.l.2c), and is used in the following in the form

96

Introduction to Photon Communication

Ro = -

~n log (J

0(

{Ri }) )

E, {Pi};

(2: ~k6 2:

= - ml n log

k,l

(11.3.3a)

";PkiP1i)

i

where S is the convex set

=1

Vj, Pi ~ O,Tr (Pi)

~i ~ 0, 2:~i

S=

=1

i

Vk,

(11.3.4)

Rk ~ O,2:Rk = n k

We follow the method [20] and also refer to relevant recent work

[31] to prove that the cut-off rate is maximized by a p.o.m. of rank one. The proof can be sketched, based on the following properties of the function operator J 0

(i) For given ei and {Pi}, J o ({Pi}, with respect to the

{Rd)

is a convex U function

Rk

(ii) Substitution of two arbitrary p.o.m., for example,

RN-l +RN-2 with R~ = Rk for j

E

[1, N -3] leads to R~ :s; R o.

(iii) For a 1id Hilbert space of dimension

'rk Irk) (rk I, such that rk :::; 1, L:k rk

RN- 2

d,

the solution is

Rk =

= d, Ilrk I = 1.

(iv) d2 ~ N ~ d. The result (iv) means that the information rate can be larger than log d which is the maximum amount of information derived from a classical consideration.

II. Performance Criteria

97

Let us apply these results to an important example: the input states are M -amplitude modulated coherent states

I 21 - M

M -1

{Pl},=o = J-t

+ 1)(J-t 21 M - M + 11 _ 1 =

M _ 1

with J-t E lR+ and (II.3.3,4) are calculated with

(11.3.5)

la,)(ad

Rk

as projectors.

The principle of the calculation is based on the well-known method of minimization of a quadratic form. In fact, introducing the matrix T whose elements are

tkl

=

yIiiki, it is clear

that' 11.3.3a takes the form ~ min (S+T T+ S). The solution of S

--

L = mjn5+GS is obtained by means of usual Lagrange multi-

pIers. The G

= {(a,la m )}

is the Gram matrix of the M 2 scalar

products of Eq. (11.3.5), and S is the convex set (11.3.4). For discrete input states and continuous ouput (e.g. measurement of the operator

at

(see 11.1.30)), the expression of Ro takes the

form

(II.3.3b)

Therefore _ uk R o - -log ( det G ) such .t h atv, Ekl wkl

" qk=L..JWkl> I

0

where det G denotes the determinant (> 0) of the definite and nonnegative matrix, Wkl are elements of G- 1 • For one or more than one

(m, n, ...) such thatqm,n,...

~

0, it is arbitrarily imposed em,n, ...

A detailled analysis is given in [28].

= o.

98

Introduction to Photon Communication

2

Ro (nats)

M 5 4

3 1 2

average number of photons

0.0 '----

....-.-.._ _

0.1

10

1

S

--.1.---1

-----1

100

Fig. 11.2. Cut-off rate versus the average number of coherent signal photons S == IJlI 2 , for a noiseless channel : M amplitude-modulated coherent states with M=2,... 5.

From the previous figure, it can be noticed that for S ==

ft2

~ 1, it

is no interest encoding the source with M > 2. Now, we would like to treat the case M == N == 2 uSIng the method pointed out in Sect. 11.2.1. We set p(111) == p == cos 2 8,p(212) == q == 1 - cos 2 (B

+ 8)"

== (aolal) == cosB and we

calculate the cut-off rate

R~2)(6,O) = for given

a~

B

~

log

2 1 + Vp(1 - q) + Vq(l - p)

I.

We seek 8* that optimizes R o. There is a

symmetrical axis 6 =

f

so that 6*

=f -

(11.3.6)

£which is the value that

minimizes the probability of error in detection and characterizes the o.d.o., as is already seen from the geometrical interpretation (cf. Fig. 11.1).

II. Performance Criteria

99

For M == 2, N == 3, a strategy that uses a third projector located in the plane defined by the two input pure states leads to

a degradation of performance R~3\ 8, B) < log(2), V8, B as can be seen by using the generalized measurements ilri)(ril, such that

Eilri)(ril == ll, (rilrj)

hij,i

=1=

min Pe is obtained for 8* = 6

=1=

j == 1,2,3. Notice that the

f - £,

where

i s

B S

2 1r. 3

We

set p(111) == p(211) == p == ~ cos 2 h,p(112) == p(212) == q == ~ cos 2 (B

+ 8),p(311) = r =

2

sin 8,p(312)

= w = sin2 (B + 8),

we

then calculate

R(3)(8B)-log o

,

-

1 + JrW +

vip( 1 -

2

q - w) +

viq( 1 -

p - r)

(11.3.7) This result can be generalized for arbitrary M, N in a particular case [15], for which we have

p(jlj) == 1 - (M - l)p - (N - M)r, p( k Ij) == p j, k == 1, ... , M == r k == M + 1, ..., N

k =l=j, k =1= j

It is simple to calculate M

(N)

Ro

= log D(N , M ,r,p)

D(N,M,r,p) == 1 + (M -l)(N - M)r +2(M

-l)VP(l- (M -l)p - (N -

M)r) (11.3.8)

which can be applied to M -ary

chann~l with

an erasure symbol.

Analysis of Eq. (11.3.8) versus Nand M for given p show that for

100

Introduction to Photon Communication

a special case of N == M and p == r, communication with two real amplitude coherent states of weak intensity

/12 ::; 10 8 M, 2

degrades

with increasing M [28]. However, for M == 2, N

= 3, performance of a binary symmetric

erasure channel with classical detection and optimized threshold

= N = 2 [5,

can be made higher than those of M

p. 400] and we

also may calculate the mutual information and then the capacity

(11.3.9)

where a 2 and b2 =

~2

(J1 + (M -

1}y -

and error probability respectively (a 2

, ==

e-p,2 T ,

vr=-:r)

+ (M -

2

1)b2

are the exact

+ € = 1), and

p is the real amplitude of the light detected within a

time interval T. For M

R=

---t

lo}M

(X),

we use once more the method of Sect. II.l.2b, where

is the rate of information and u = ~ to prove that the

capacity is such that

o < u < 1, u

2: 1,

if, 2 ,,2 brvb2

rv

C

rv

(1 - 1') log M

C

4 '

rv

log M

--+ 00

--+ 00

and the cut-off rate will be given by

R o = log

M

M - 1 + € - (M - 2)a 2

+ 2(M -

~

1) a2 b2

plotted in the next figure for several values of M.

(11.3.10)

II. Performance Criteria

101

Ro

8

7 (nats) 6 5 4

16

3 2

4

o o

~---L-_--i.------I_-..I--_-'---.l._--I.-------'

2

3

4

5

6

7S

8

Fig.II.3. Cut"':'off rate performances for M -ary orthogonal signal coding quantum channel for M = 4, 16, 32, 64, 128 versus S for f. = 0, (a 2 = 1- (M -1)b 2 ), versus the average input signal photon number S = J-l 2 T. The asym ptotic curve is quoted S and is given by R o= - log '/= S. It is observed that we have Ro(M, S) ~ S, \:1M, S.

II.2.3c Rate-Distortion As in Sect. II.l.2d, the joint input-output probability, the input probability and the output probability are rewritten here in terms of the

{pj}input density operators and the p.o.m. {Rk}

7rjk = TrejpjRk Pj

= Ek 7rjk = ej

Tk = E j 7rjk = Tr pRk P= E j ejpj = Lj ejl?jJj)(?jJjI, (ej ~ 0, Lj ej = 1, TrPj = Lk q(klj) = Lk TrpjRk = 1, Vj, (Lk Rk = ll)

1, Vj)

(11.3.11 )

102

Introduction to Photon Communication

We then need to define the measure of distortion. It is taken here again as L1jk = 1 - 6jk (see Sect. II.1.2b), where we assume that j, k = 1, ... , M. Its average value is given by

(11.3.12)

As previously, the mutual information is

=

L 1rjk (log 1rjk -log rk) - L ej log ej

(11.3.13)

j

j,k

To give (11.3.13) a different form, we notice

S(it) = -Tr (itlogit)

= - L1rjklog1rjk jk

where 1rjk

= (1/Jj, xklitl1/Jj, Xk), it = {ejpjRk}, (L,jk 1rjk =

1).

Therefore Eq. (11.3.13) can take the form

I(jlk) = R(S) + Tr (it (log it -logRk)) where H(:5)

(11.3.14)

= - L,jejlogej.

Then the rate distortion can be rewritten as solution (Rk) to the problem of optimization for given :5

R(D)

= nEn [H(5) + Tr ( ii (log ii - log Rk) ) ] Ric

D

= Tr (itJ)

(11.3.15a)

II. Performance Criteria

103

that can be solved by using the standard Lagrange multipliers method. Here, because D

= Pe ,

it is simpler to make use of Eq.

(II.1.26b) and show that the expression of the p.o.m. II =

{Rd

is

such that (II.3.15b) given

"'" Kk

~j ~/pje8i1j"/lj and

/lj

=

~k Wjk, Wjk

=

{A-l }jk.

From Eq. (II.3.l5b), it follows that (II.3.15c) where dM

=

MM"l and D is simply related to the scalar product

of the input states (see II.2.2c). As we consider here pure states, the inverse of Pexists for non commuting pairs

[Pj, Pk]

=1=

0 (Sect.

I.l.la). We conclude this section with some explicit expressions of the rate distorsion functions computed for different receivers in a binary channel: E = (e, 1 - e). Actually, except for the o.d.o., the mutual information is not minimised subject to a given distorsion measure. (i) For example, given

Rk = N and

for 0 ~ D < 1 -

e, we

obtain [19]:

IN(D)

= Dlog(D)-(l-e)log(l-e)-(D+e)log(D+e)

By noting that for

-(I {

(II.3.l6a)

e, D ~ ~, we have

0 log(l -~) -

(D +~) log(D +~) :::; -(1 + D) log

DlogD ~ D(D -1) -(1 + D) log(l + D) ~ -D(D + 1)

l

1D

(II.3.16b)

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Introduction to Photon Communication

A very simple upper-bound can be written as

IN(D)

~

(1 + D)log2 - 2D

which appears reasonably good for ~ ~

(11.3.16c)

t. A lower bound seems

more difficult to obtain. We found IN(D) ~ -~

+ ~ + (l~e )Oog~.

This bound is rather poor but appears useful for some values of I

~

rv

0.2. (ii) As a second example, we consider the homodyne opera-

tor

Rk

==

X for which the measured signal is a continuous variable: Ix(D) = H(p) - H(r) for 0

~

1

D < 2"

(11.3.17)

where H(p) == p log(p) +(1- p) log(l- p) and r == p+ ~(1- 2p). The probability p == Prob(k so that p

== 0li

= D = t(1 - Q(

== 1) == Prob(k ==

If))

Iii

== 0), is given

where Q( u) is the error function.

(iii) The third example is the o.d.o. completely defined by

A=~; p = rrAhj

h=1-e- s ; A=VAh+(l-,.>")2; 1J=A+ 1 ;>"

q = 7]t~hj ro

= ~(1- p) + (1- ~)(1- q)

D == ep + (1 - e)(1 - q) Io(D) = ~ ((1- p)log l;}

+(1 -~)

+ plog~)

((1- q) log 1::afJ. + q log ~) (11.3.18)

where we set S == 1("pll"po) 12 and l"po), l"pl) are the pure input states. Because the distorsion criterion is of a probability of error type, it is clear that Io(D) == R(D), R(D) being the classical rate distorsion function (11.1.26).

II. Performance Criteria

105

o.? r-~(7=--:):-r-----r---r--.,---.,-----,-------r--r-----,

D

(oats)

0.6

cs ss

0.5

TS

0.4

s

CS

1&'0

I

2

J

4

SS

0.3

o.d. .

0.2

D

0.1

o o

L.----L-_-L.---.L_--L-_..L-------L-_--L------J_.=---.;:::w

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Fig. 11.4. Mutual information I(D) for binary channel in the case of

a priori probability x = 0.5 for fixed probability of error Pe = D for photoncounting channel (N). Notice that the three types of light, coherent (C8), squeezed (88) and thermal (T8), yield same I(D) although the Pe are different. The (o.d.o.. ) curve calculated for optimum detection operator with eqs. 11.3.18, corresponds to the rate distorsion function given by eqs. 11.1.26.

Now, due to the fact that the rate distorsion is a strictly monotonic decreasing function and as IN(O) D~

= R(O) = H(e)

and because

> DrJo ( DMthe maximum value of the average value of the

distorsion for a receiver r), we can prove that IN (D )

~

R( D ).

Therefore, it is observed that higher complexity encoder than the photocounting scheme is required to achieve a performance closer to the R(D). This is due to the fact the photocounting channel is of Z-type that is to say a non-symmetrical one. A more detailed version of these results will be published in a future work [36].

106

Introduction to Photon Communication

II.2.3d Channel with Thermal Noise: Approximate Results Let us consider M pure coherent states described by the density operators

Pi

==

IJli) (Jli I where Jli is the complex envelope of the in-

We are also given the scalar products lkm

ei,

0, ... , M - 1).

== (JlkIJlm).

For noiseless

channels, the orthogonal projectors of rank one the resolution of identity L~~l fij

(i

==

put field given with equal a priori probability

= nM

fij = IWj)(wjl with

are the optimal mea-

surement operators to minimize the probability of error [16, p.98]. For binary channels with thermal noise of weak average photon number No, in order to solve the detection equation (PI -Apo)ITJ) ==

11111), some approximate results have been established in terms of the function F(f3*) = e

l

.8t (1311]). The linear equation satisfied

by F(f3*) is expanded in v =

l.fNo

and it is shown that up to

the first order, the false-alarm and detection probabilities can be approximated [9]. Another method, based on the modification of the resolution of identity [16, p. 184] and of simple interpretation, is developed below. In the presence of thermal noise, the density operators [33] are first expanded under the P( a)-representation [1,2]

(II.3.19a)

where la)(al = D(a)IO)(OIDt(a). These states are then projected on the coherent states {IJli)} in order to evaluate

II. Performance Criteria

107

(JLkIPiIJLl) = 1 +1 No 2

2

2

No * + JLkJLi + JLiJLm - IJLi 1 IJLk 1 IJLi/ ) exp ( 1 + No JLkJLl 1 + No - -2- - -2(11.3.19b) Therefore, the measurement process involves a resolution of identity on the infinite dimensional Hilbert space as required by (11.3.4) that is a difficult problem to solve. However, some useful bounds are easy to obtain. In fact, as was done for the detection problem, we assume that the new measurement operators are approximately given by

(11.3.20a, b)

Now, it is useful to expand the detection operator on the input states to read M-l

IWi) =

L aljlJLj)

(11.3.21a)

j=O

so that for the probability distribution

Pij = Tr (Piiii) = (wjIPilwj)

+~

(1 -~\WiIPiIWl)) i=O

(11.3.21b) where

(wmIPilwm) =

L amla~k(JLkIPiIJLl)

(11.3.21c)

ki

As shown [26], the function J0 defined by Ro upper bounded in the following manner

=-

min log (J 0) is

ei

108

Introduction to Photon Communication

L ekell JJ Pk(P)PI(P)d pl

M-1

2

Jg 2:

(II.3.22a)

k,l=O M-1

M-1

L

Jg 2:

ekel Tr [fhpl]

+ L e~

where Pk(/-l)

(II.3.22b)

k=O

k,l=O, k:¢:l

2

=

_1_

1rNo

exp (_la-ILk 1 ) . No

Demonstration of these bounds is based on the P( a) representation and the spectral decomposition of the density operator. N-1

M-1

L

J =

ekelhl,

hi =

J 2: I J

(i)

Pi =

(ii)

hi

Tr (Pill:) Tr (Pill:),

2

Pk(p)lp)(pld p JPk(P)PI(p) L(plll:lp)d2 pl n

IJ J Pk(P)PI(P)d pl 2

=

(i)

L

n=O

k,l=O

k = £,

-+

L ll: = 1, hk = 1

(II.3.22a)

-+

n

(ii)

k

-# £,

J =

L e~· k

Pi = Lj Ajlw~k»)(w~k)1

" \. n(kn) Jtj n(kn)* Tr ( ""'Pk-""R""'n-"") - L....Jj A]Jtj 2 0,

(Q~kn) = (w~k)Ill:lw~k»)), Jkl2

LTr (Pill:) Tr (Pill:) = n

(iii)

(i)+(ii) ~ (II.3.22b)

JTr(PiPi)

II. Performance Criteria

109

Both bounds are useful, the domain of their interest depend only on the function f(S, No) == max {t2

(e- Jo , VI +1 2Noe- Vlt2NO), where

== S. Let us illustrate the calculations by an application to the noisy

binary channel. It is emphasized again that the equation of optimization is not solved. Only approximate results are calculated. We find the transition probabilities for this binary symmetrical channel (OOK modulation

{to

== 0, {tl == {t,,2 == e- S ), e.g.

(11.3.23)

{

s

where the expressions (DlplID) = HlNo e- l+No ,(J.LlpllJ.L) = HlNo are

calculated for states with real amplitudes. To obtain (11.3.23), the following relations are used

(wo IPi Iwo)

(wllpi!wl)

= 12~~ (po IPi Ipo) =

12 (po IPi IPl)

2

+ 2Ll2 (i + Ll) (Pllpi Ipl)

2 12~~ (po IPi Ipo) - 12 (po IPi IPl) + 2Ll2(i _ Ll) (PlIPi IPl)

where L1 2 == 1 -

,2.

The cut-off rate is therefore easily derived

R o == log

2

I

+ 2Vpoo(1 -

2

Ptt)

== log -----;== I

+ VI -

f32

(11.3.24)

110

Introduction to Photon Communication

In order to compare with the bounds·, we prove

(11.3.25a, b)

On the other hand, as shown in [28], when the a priori probability are taken uniform, the performance are suboptimum. Furthermore, the operators used to compute the R o are derived from the o.d.o. which is optimum only for detection criterion. It is then clear that the expression given by (11.3.24) is in fact a lower bound denoted

Rb

and calculated as

Rb = log 1+~.

Now, depending on the value of S, the upper bound is given either by (II.3.25a) or (II.3.25b) as will be seen in the following expressIons.

Rg = log (Jg) = log (

Rg

= log (Jg) = log (

1+

2 1

Y'l+No

8_)

__ e l+No

2_-L)

1+e

4NO

These bounds are plotted in the next figure for No noticed that

-

S ;S 0.02

R~ ::; R o ::; R8

s~

R~ ::; R o ::; Rg

0.02

N o ;5 0.02:

(11.3.26b, c)

R&::; R o ~ Rg

=

0.1. It is

II. Performance Criteria

111

0.7 ,..-------r------r------,----:====:::;;::::I Ro (nats) No == 0.1

s

0.0 L-_==:::=:...&-_-==:;;;;..--"'--_ _ 10- 3 10- 2 average number of photons 10 -----I

---J

Fig. 11.5. Bounds for the cut-off rate versus the average number of coherent signal photons S = 1J-L1 2 and for a noisy channel of a weak average number of photon thermal noise N o=O.l.

11.3 References [1] [2] [3] [4] [5] [6] [7] [8]

R.J. Glauber: Phys. Rev. 131 2766 (1963). W.R. Louisell: Radiation and Noise in Quantum Electronics (Mc Graw Hill, New York 1964) Chapter VI, pp. 220-252. R.G. Gallager: Information Theory and Reliable Communication (Wiley, New York 1965) H. Takahashi: Advances in Communication Systems 1 (Academic Press 1966) 227-310. J. Wozencraft, I. Jacobs: Principles of Communication Engineering (Wiley, New York 1968). H.L. van Trees: Detection, Estimation and Modulation Theory (John Wiley & Sons, New York 1968). Part I. A.V. Balakrishnan: Communication Theory (McGraw-Hill, New York 1968) Chapter V, pp. 192-215. A. Spataru: Theone de la Transmission de l'Information (Masson, Paris 1970)

112

Introduction to Photon Communication

[9] [10]

R. Yoshitani: Jour. Stat. Phys. 2, 347 (1970). C.W. Helstrom, J.W.S. Liu, J.P. Gordon: Proc. IEEE 58 1578 (1970). T. Berger: Rate distortion theory, (Prentice Hall, Inc., Englewood Cliffs, N.J. 1971). A.S. Holevo: Problemy Peredaci Informacii 9 31 (1973). R. S. Kennedy: Res. Lab. Electron., M.I.T., Cambridge, Quart. Prog. Rep. 108 pp. 219-225 (1973). A.S. Holevo: Problems of Information Transmission 9 117 (1975). H. P. Yuen, R. S. Kennedy, M. Lax: I.E.E.E. IT-21125 (1975) C.W. Helstrom: Quantum Detection and Estimation Theory (Academic Press, New York 1976). E.B. Davies: Quantum theory of open systems, (Academic Press 1976). R.J. McEliece: The Theory of Information and Coding (AddisonWesley Pub. Co., Reading (USA) 1977). C. Bendjaballah: in Coherence and Quantum Optics IV, ed. by L. Mandel and E. Wolf (Plenum Press, New York 1978). p. 241. E.B. Davies: I.E.E.E. IT-24 596 (1978). J.R. Pierce: I.E.E.E. COM-26 1819 (1978). J. H. Shapiro: I.E.E.E. Trans. Inform. Theory, IT-26 490 (1980). R.J. McEliece: I.E.E.E. Trans. Inform. Theory, IT-27 393 (1981). J.L. Massey: I.E.E.E., COM-29 1615 (1981). A.S. Holevo: Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam 1982). M. Charbit: Information performance for communication channel with thermal noise (LSS Internal Report 1988). W.H. Zurek: Phys. Rev. A 131 4731 (1989). C. Bendjaballah, M. Charbit: I.E.E.E. Trans. Inf. Theory, IT-35 1114 (1989). R.M. Gray: Entropy and Information Theory (Springer-Verlag, Heidelberg 1990). W.H. Zurek: Complexity, entropy and physics of information, ed. by W.H. Zurek, (Addison-Wesley, Redwood City, USA 1990), Vol. VIII. K.R.W. Jones: Annals of Physics, 207 140 (1991). T. Cover, J .A. Thomas: Elements of information theory, (John Wiley & Sons, New York 1991). A. Vourdas: Phys. Rev. A, 46 442 (1992). M.J.W. Hall, M.J. O'Rourke: Eur. Opt. Soc. preprint.

[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

[31] [32] [33] [34]

II. Performance Criteria [35] [36]

113

T. Uyematsu, C. Bendjaballah : I.E.E.E. Trans. Comm. to appear Apri11995. J.M. Leroy, C. Bendjaballah : in preparation.

III. Direct Detection Processing

The method called direct-detection and its practical importance have been briefly analyzed in the Introduction from a sensitivity point of view. As seen in Sect. II, it is only suboptimum, as regards to the quantum detection performances which have been rigorously defined in [35],[37],[39],[58]. However, in many practical cases [38], it achieves interesting performances. These receivers utilize the detectors e.g. photomultiplier tubes, avalanche diodes, p.i.n, · · · in order to measure directly the timeintegrated low-power optical signal. In the treatments of the direct detection of the quantum limited signal, it is proved that the observations of the photodetector output constitute a random point process denoted "r.p.p.".

This r.p.p., analysed from the quantum mechanics [3], quantum stochastics [24], and classical [22] points of view, is of the doubly stochastic Poisson process type: conditionally to the given intensity (definition of the intensity as a random function of time for the point process treatment or as a stochastic operator for the quantum stochastic calculus, is always possible), the process is an inhomogeneous Poisson process for which there are many results. Mathematical proofs of these results are demonstrated using the calculus of probability in [34],[41],[43],[55],[70] to quote references that can be related to optical detection. Hence, these results are very important because they induce the necessary relationship between the input and the output for the detection of light-field. Often the basic typical model is the pure Poisson counting process [5],[11],[12].

III. Direct Detection Processing

115

In optical communication, it is adequate for representing the detection process of a source of information of constant intensity. . So far, the non-deterministic behaviour of the output process was supposed to be a result of the photocathode response. Now recent theories prove that the randomness is· related to the source fluctuations (vacuum fluctuations) and the photodetector can be considered as being of a deterministic nature. The random fluctuations in the electron multiplication process is of no special importance for the present purposes. In this section, we first recall the basic definitions and properties of the r.p.p. from the different points of view mentioned above. We then demonstrate the main results relevant to detection and information performances for direct detection.

111.1 Randolll Point Processes Theory In the following, we present elements of the quantum and classical approaches to the characterization of the random point processes. Basic definitions and the main results will be briefly reviewed.

111.1.1 Quantum Formulation Let us consider the electromagnetic field given by a collection of harmonic oscillators. The Hamiltonian is given by

Ii =

L AWl (atal + alat)

(111.1.1)

I

where the operators at and az are

(111.l.2a)

116

Introduction to Photon Communication

(111.1.2b) The solution of the Schrodinger equation for the electrical field operator can be written (initial time 19 0

= 0)

(III.1.2c) where V, the volume, is introduced in order to handle the discrete sum. To simplify the presentation, we will restrict our study to a single mode light field. It is convenient to write the electric field operator in the form [2,31]

(111.1.3) The correspondence between the classical and the quantum formulations, as established by Glauber [2] for a given density operator of the source

where Eel

p= l{al})({al}l, reads

= 2i ~e V!hii Wal

~ ~

ei ( k I r

-wlt?).

A simple model of pho-

todetection consists in considering only the dipolar interaction

- PE(-:t, {)) for a given polarization between an atom of the photodetector and the electric field. The calculations to obtain the probability of detection of a photon are briefly summarized as follows [2,3]. We start with the matrix elements of the U

ca, 19

0)

time operator

evolution between the initiall
III. Direct Detection Processing

Let Weg be the frequency of the atomic transition l
l7Pi)

--7

l7Pt)

--7

117

l
be the field transition.

The probability of the atomic transition at (), the field being in the

initial state defined by the density operator

P({)) = Tr

[il L

il, is given by

1(4)e,¢fIU(19,190)I4>g,¢i)12]

{ltPJ)}

=

;2 Jl d191d192F(Weg,(192 -19d) x Tr [ilE(~, 19 1 )- E(rt, 19 2

)+]

(111.1.4) where F (w eg , ({)2

-

19 1 ))

= exp (iw eg (19 2 - 19 1 )) 10"eg 12 ·

Using the expansion (III. 1.3), only terms of E(~, 19)+ will contribute in (111.1.4), because they conserve energy in the ionization process. The probability density to register a photoelectron in the time interval

e = [()o, ()] is therefore given by

Tr .17 Tr

[il E(~,19d-E(~,191)+]

[il E(~,19)- E(~,19)+] (111.1.5)

where T (19) is the Fourier transform of the spectral response of the photodetector and ", is a coefficient characterizing the detector.

118

Introduction to Photon Communication

111.1.la Time-Occurence Probability Density For a generalization of (111.1.5), let us consider n time intervals along the time axis and define the event F n for which one photon is measured in {19 1 , 19 1 any other in {19 n ,19 n

+ d19 1 },

another in {19 2 , 19 1

+ d19 2 },. · .,

and

+ d19 n }. The probability density of the event

F n is called time-occurence probability density and is given by Prob {[dN (19 1 ) == 1] [dN (19 2 ) == 1]··· [dN (19 n ) == I]} = Tr

where

{RO [M (19 1 ) M (19 2 ) " , M (19 n )]}

(III.1.6a)

0 is the normal ordering operator. From (111.1.5), the oper-

..-..

ator M is

M(19 i ) =

L JJdrtdrtV/11,/12 (rt, rt) Ill,1l2

X

(III.1.6b)

where III and 112 are the polarizations of the potential vector and

V /11 ,/12

(rt, rt) is the detector response.

111.1.lb Counting Probability In the following we omit the spatial dependence on -r* and restrict ourselves to time properties of the field of a given direction and polarization. Here we define the basic operators and functions needed for the counting statistics. First, the counting operator is defined by (III.1.7a)

III. Direct Detection Processing

119

Therefore, from the previous equations (111.1.5)-(111.1.6), we may set

(111.1. 7b)

For a time independent, single mode-free field with

M = "lata the

generating function takes the form

G(s)

= Tr [RO {exp (-s"lTatan]

(III.1.8a)

For this specific case, the coincidence probability is defined by Prob {[dN (19 1 ) = 1] [dN (19 2 ) = 1] ... [dN (19 n ) =

I]} (III.l.8b)

= ("lTtTr [R8{(a t at}]

and from (111.1.7b), the counting probability density is given by

. . . . . . . . . {("lTlit Prob[NT = n] = Tr [RO n! For a stationary field

[iI, Rl

at

exp( -"lTa t a) }] (III.1.8c)

= 0, and setting

R=

Ln Rnnln)(nl

we obtain

00

~

(111.1.9)

,

m

m.

=~"l(m-n)!

R mm

120

Introduction to Photon Communication

Remark: It should be emphasized tllat the number of photons absorbed by the detector canl10t be larger thall the total number of photons present in the field. This question has been studied by Mollow [16]. We refer the interested reader to the work of Davies [24] and Srinivas and Davies [54] for further details. Instead of (III.I.IOa), the photoncounting distribution for a single mode-free field will be given by

from which it is easy to prove that E[n]

~ Tr

[Rata]

as required.

Notice, however, that in most cases this correction is not significant. In (64], it is demonstrated by solving the equation for the time evolution of the annihilation operator

a in the Heisenberg repre-

sentation, that the probability of detecting a photon is of a Poisson type (III.I.8e).

111.1.2 Quantum Stochastic Formulation Let, at the initial time s, the state of the field on the space denoted IC, be given by a coherent state la), (j == la)(al. The field operators are the usual creation, annihilation and number

at (s), a( s), Ns

as already defined. According to Barchielli and col.

(see [73] and references therein), adapted process operators (after interaction) are defined by integrals of these operators --

A(t)

= 10t

def

a(s)ds

III. Direct Detection Processing

121

The rules of the quantum stochastic calculus established by Hudson and col., are summarized in [73]. We just recall here that the

derivative rule is d(Nt · it (t)) = dNt" it (t)

+ Nt" dit (t) + dit (t) ..

ii

in a separable Hilbert

Under the self-adjoint Hamiltonian

space 1i, the system (defined by the density operator ji) in interac-

tion with the field, evolves with a unitary operator U(t) E 1i ® Je, given by the stochastic Schrodinger equation

{

d~(t)= [-Rt.dA(t)+R.dAt(t)-~Rt.Rdt-iiidt]U(t) U(O) = n

where

Ris the system-bounded operator on. 1i, which is of interest

here. Thus, at time t, the density operator of the system can be written as

PC t)

= Tr JC [ U(t ) (p ® (j) U(t) t]

Let us now consider the device we have already called the photoelectron counter, which realizes the measurement of the observables, number and intensity operators, as follows

Nt,t?o = U( t) t (it at (s )a( s)ds) U( t) let) = it dN

(III.I.lla, b)

s

We introduce the characteristic operator (analogue of the characteristic functions (III.I.8a) and (III.I.14a) below)

122

Introduction to Photon Communication

that obeys

d~t(U) = ¢t(u) [exp (iu -l)dNt ] {


=n

More general definition uses a time function k( s) instead of a constant u. Therefore the characteristic function of the point process is given by

which yields the probability distribution of a compound Poisson type. In the following, we will be working with a classical approach which is, in our opinion, more appropriate to interpret the questions of physics.

111.1.3 Classical Formulation The £, (iJ) incident is a quasi-monochromatic light field which is considered as a real, random, stationary function of time [4,15,23]. Its Fourier transform is written in the form e(w) == a (w)e i 4>(w). The complex function of time Z (iJ), called the analytic signal is obtained from the electric field by the transforms

(III.1.12a)

£, (iJ) == ~e{ Z

the

~e{Z (iJ)}

and

~m{Z (iJ)}

(iJ)}

(III.1.12b)

being related Hilbert transforms.

III. Direct Detection Processing

123

In (III.l.12a), Wo and·tP (w) are the central frequency, and the phase shift, respectively. We denote as quasi-monochromatic a light field for which the bandwidth L1w is such that L1w ~ Wo. The emerging current intensity from the photodetector will be

IZ (19) 12 • Although the quantum definition of the intensity operator is possible as J = 8{E+E-} or as (III.l.llb),

given by J (19) =

the classical definition requires positivity of the intensity which is difficult to verify for some cases of the squeezed light. Let

,J(19

1,

19 2 ) be the normalized second order time correlation

function of the random function J (()). The coherence time of the stationary random function J (19) is given by where u

= 1'19 1 -

= 2 10 I, (u) Idu 00

Tc

19 2 1.

Similarly, for the random function of space, we call the coher-

ence length r c Uo

= 2 It)

I'J (-~Iuo) Id~ where ~ = 171 - 711 and

= 119 1 - 19 2 1. Now for a fixed position of the ·2D-photodetector of a spatial

area A which is considered as to be of a constant gain, infinite bandwidth and without internal background noise, the instanta-

neous intensity becomes p (19)

= 17 J (19)

with 17

=

::a

and 17 the

quantum efficiency of the detector. In the following, the spatial

properties which are especially important for image processing or space localization of parameters [32,33,74] are not considered. In this section, we set

'1J

= 1.

124

Introduction to Photon Communication

111.1.3a Main Properties of a Doubly Stochastic Poisson Process It is useful to briefly recall the definitions and properties of the pure Poisson r.p.p. denoted "P-r.p.p." as follows (i) Let ({ 19}n; n

2:: 1) be a set of instants on the time axis.

Let {Ti == 19 i - 19 i-l }i=l be independent random variables identically distributed (i.i.d.) with probability density p( Ti == x) == p e- Px , for x > 0, p being deterministic, constant and>

o.

(ii) Let T E lR+ and NT be the number of points occuring in T,

defined such that NT = En~l 8 (T - t9 n ). The {NT} is the counting random function.

2:: 1) be the set of parameters p > 0 defined in (i). For 0 < T1 < T2 < ... < Tn, the random increments

(iii) Let ({19} n; n

NT1, NT2

-

N T1 ,···, NTn

-

NTn _ 1 are independent. More-

over, for T' < T, n E :IN, Prob [(NT - NT') == n] ==

Ip(T-T')r n!

e- P(T - T') •

It is simple to demonstrate the reciprocal proposition. (iv) Let W n == E~l Ti. From the above definitions, NT ::; n iif W n +1 2:: T, T > 0, n E IN, so that the equality Prob[NT < n] == Prob[Wn > T] is the relationship between

counting and interval distributions. We begin by establishing the main elements for the generalization of the -above definitions to conditional "P-r.p.p." [22,27,34]. We consider that the output of a photodetector receiving an optical signal of very weak power, is a set of narrow pulses over the time axis. These pulses correspond to the instantaneous emission of a

III. Direct Detection Processing

125

photon consequent to the absorption of a photon. This sequence of time instants {'l9i} constitutes a r.p.p.. Now, let this r.p.p. be characterized by the counting function that is the number of {iJ i} in the time interval T: N (T) ==

IT dN (iJ), where dN (iJ) are the random increments of N (T). It can be seen from the theory of probability that the complete characterization of N (T) requires the knowledge of the n-dimensional probability distribution for each n: Prob[dN(iJ 1 ), ... ,dN(iJ n )] for every set {iJ i } ~=1. Then, the following properties characterize the process Prob [dN (iJ)]

> I == 0 (diJ)

that is, the random variable dN (iJ) can take only two values

o or 1. Therefore, Vk, dN (19)k =

dN (19).

(ii) for different {'l9 i}

Prob{ [dN (19 1 )

= 1] [dN (19 2 ) =

1] '" [dN (19 n ) = 1] Ip(19) }

== p (iJ 1 ) P (iJ 2 ) ••• p (iJ n) diJ 1 diJ 2 ••• dfJ n This is the classical form of counting probability density (III.I.8e). For photon counting purposes, let us deal with

To evaluate this integral, it is useful to start with the characteristic function

¢>N (ulp) def EN [exp {iuN (Tn]

= exp (iU

£

dN (19))

126

Introduction to Photon Communication

The interval of length T is conveniently divided into equal subintervals 8j such that with the probability (111.1.13)

1 - p ({) j) 8j

with the probability Hence cPN(ulp)

=

exp (iULjL1NC19j)) which becomes, taking

into account (111.1.13)


= 11 {l+(ex p (iu)-l)P(t?j)Dj} j

~

11 {exp({exp(iu) -l}p(t?j)Dj) } j

= exp [L (exp (iu) -

1) p(t?j) Dj]

j

---t

6j

~O

exp [( exp (iu) - 1)

iTf p ('11) dt?]

Finally

(III.1.14a, b)

In view of (111.1.14a,b) and (III.1.7b), we can conclude that these approaches, briefly presented above, are "equivalent" at least for the one fold statistics.

III. Direct Detection Processing

127

Remark: In order to take into account the decay of the field state after interaction with the detector, it is sufficient to replace exp (- IT P(19) d19) in Eqs. (III.1.14a,b) by its first-order expansion 1-ITP(19)dfJ.

III.l.3b Photon Counting and Time Intervals Distributions For the rest of this section, we will adopt this formulation which is more suitable for comparison with the experiment. In fact, treat,ing the detector separately from the source makes the interpretation of the optical detection easier. The distribution given by (III.1.14b) is called counting probability density. For an inhomogeneous Poisson process, P({)) is deterministic. For a pure Poisson process, P(()) is constant. In both cases there is no need for averaging over the density of the process and (III.1.14b) takes a simple form. As pointed out previously, instead of using only the counting process {N (t) It E [0, T] } to characterize the occurrence properties of the r.p.p., it is sometimes useful to use the time-occurrence

process

{W (n; t) In E

[1, oo[}. In general, methods based on count-

ing and time-occurrence distributions are quite different. This can easily be seen: the basic relation between these two distributions is given by (III.l.2b (iv)): N (T) ~ n iff

o; n

W (n

+ 1; t)

~

t t >

E [1, 00[. To calculate the needed distributions, it is conve-

nient to start with the probability of the n-occurences at {{) i} for each n

Prob(t9 1 ,t9 2 , ••• ,t9 n lp)

= p(t9t}p(t9 2 ) ••• p(t9 n )exp (-1~ P(t9)dt9) (III.l.15a)

128

Introduction to Photon Communication

The marginal distribution is obtained by taking the average value over p

Prob(19t,19 2, ... ,19 n ) = E p [gp(19 i )eXp

(-iT P(19)d19)] (III.l.15b)

The distribution given by (III.l.15b) is called time occurrence probability density. For p(iJ i ) = Pk(iJ i ), we denote Pk({f}}i) the expression (III.l.15b). The characteristic function and the counting probability distribution given by (III.l.14a, b) are the functiol1S which are sufficient for the treatment of the r.p.p. in the context of this section. Other functions of interest can be defined as well. From (III.l.15c) and

N

G (8; T) = Ep [ exp ({exp (-8) - 1}

iT P(19) d19)]

(III.. 1. 15d)

which are the intensity and moment generating functions respectively, it is simple to derive the counting distribution

an '-a G p (s; T) n. sn

(_l)n

p(n; T)

=

I

8=1

(III.l.15e)

Because the origin of the time axis is arbitrary, this distribution is sometimes called relaxed [21]. Generalization to M -time joint counting is straightforward, at least formally. To obtain explicit formulas, some elementary algebra is, however, required.

III. Direct Detection Processing

129

For simplicity, we continue assuming stationarity of the light field

the M-fold counting distribution being given by

Example: Let us consider a simple case of the two fold counting [17]. For M

= 2 and tl = 0, r ( n,

Tl

= 0;

t2

= T;

T2

= T,

T ..) _ Prob{[dN (0) = 1, [N (T, T + T) ,T Prob[dN (0) = 1]

__ 1

[

T

[J;+T p ('19) dt9] n _ J:+ P(D)dD] T

- E[p] E p p(O) A case of interest is

= n]}

n!

e

= 0,

1(-1) nan {I;atGp(s;t) a } Is=l,t=T n! as

r(n,T;O)=q(n;T)= E[p]

n

Because the origin of the time measurement is a point of the r.p.p., this distribution can be called triggered [21]. The time intervals distributions can be calculated similarly. We can easily show

'V n

=l(t)=E p [p(t)ex p

(-l

T

tp

+

(t9)dt9)] =-~Gp(s;t)IS=l (111.1.15 f)

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Introduction to Photon Communication

In the theory of the r.p.p. (see for example [11,12,34]) this is called the waiting time distribution. Similarly, one obtains

(111.1.15g) This is called the life time distribution. It is interesting to notice that for exp (- J;+t p('I9)d'l9)

« 1, W n =l(t;ti = 0)

rv

/'I(t) where

!I(t) == Ep[p(O)p(t)] is the time correlation function of the light intensity. Generalization for higher n is shown below. Let the moment-generating function written without taking into account the stationary properties of the light field

We first make two partial differentiations with respect to t l and t 2 •

Then we take into account the stationary properties of the field, so t;hat the functions only depend on t == t2 - tl. We arrive, for n > 1, at

III. Direct Detection Processing

131

which lead to

Now we seek the relation between w (t; ti

= Oln) = w (t; Oln) and

pen; t). We first notice that

8 &tP(n;t)

I = (n -1)!

[

E pet)

__ 1 E [p (t) (

(n)!

t

Jo

(

iot

p(iJ)diJ) n-l e- fo

t

p (iJ) diJ) n e -

f:

p(t?)dt?

]

P(t?)dt?]

A second derivative with respect to time shows

82 &t 2P (n; t)

= E[pl{ wn(t; 0) -

2W

n-l(tj 0) + W n-2(tj O)}

Hence m

L

82 &t 2P(lj t) = E[p] { wm(tj 0) -

W

m-l(tj O)}

1=0

so that

wn(tj 0) {

vn(t)

= E[p] E~=o E;:'o ~p(lj t)

=-

E~=o %tp(lj t)

(III.I.IBa, b)

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Introduction to Photon Communication

III.l.3c Some Specific Cases For counting processing, T is constant and N (T) == n is the random variable. Here the number of counts equal to 0,1,2, ... , n is kept constant and the time interval T == t is taken as a random variable. The following general expressions hold for a long t 00

Gp(s;t)

= Lsnp(n;t)=E[exp( -(l-s)pt)] n=O def

==

9 ( s; t) exp (- h ( s; t))

~Gp(s;t) I == E [pt e- pt ] as 8=0

== p(n == l;t)

[)8t Gp (S; t) I8=0 == -v(t) Similarly, one obtains wet)

= E[~]t2P(n = 2;t).

(i) For coherent light, the r.p.p. is memoryless, and we easily

find

p(n;t)

=r(n;T,t) =q(n;t)

=

(E[p~tt exp(-E[p]t) n.

(ii) For thermal light, it will be shown (see III.2.3c) that the characteristic function

~ 1- iu~[p]Am

In the case of long coherence time, we have

It p(iJ)diJ

Closed expressions can then be derived from g(s; t)

h(s;t) == 0, such that

rv

E[p]t.

= Hsk[p]t

and

III. Direct Detection Processing

p(n;t) == (1-

Vt)v~ ,

q(n;t)==(n+l)

(1 -

133

Vt)2 v n +1 t

Vt

(III.l.17a, b) where Vt

=

E [p] t 1 + E [p] t

(III.l.17c)

Moreover, for arbitrary r, we obtain [21] n (

r(n,tiT)=Vt

2n

- E [p]

t)

1+ lfE (T)1 1+ E [p]t

(III.l.17d)

Notice that the measurement of r(n, t; r) is an indirect determination of the time correlation function of the light .field. Recall that in the present case the normalized time correlation function of the light intensity is given by 1/I(r)1 == 1 + l/E(r)t 2 • The time interval distributions are simple to derive v

(t)

E[_p]_ - (1 + E[p]t)2

2E[p] w(t)---- (1 + E[p]t)3

(III.l.18a, b)

Notice that in order that the moments E[t k ], the calculation of the moments from these distributions requires limiting t E [0,00[. (iii) For a mixture of thermal of long coherence time and of photon average number E[pt], and coherent radiation of photon average number E[pc], the intensity distribution is

W( ) = m + 1 -(m+<;tpVph p E[p] e 0

(2 J m(m + l)P) E[p]

where I o( x) is the zero order Bessel function of imaginary argument. The parameter m = ~ stands for the ratio of the single

134

Introduction to Photon Communication

mode coherent intensity over the average thermal intensity and the average of the total intensity is E[p]

= E[pc] + E[pt].

The k-moment of the total intensity is found to be k

,

E[p ] = k.E[p]

k

exp( -

r; )

Jffi( m m+1 )k Mtll o(m) 2

,

where Ma,p( m) is the Whittaker function. It is interesting to notice

that the parameter h = ~ ~ is E [1, 2]. The photon distribution can easily be computed

(m + l)(E[p]T)n e - (m + 1 + E[p]Tt H

p (n· T) -

,

mE[p]T m+l+Elp]T

L (

m(m +'1)

------

n

)

m + 1 + E[p]T

(111.1.19) where Ln (z) is the Laguerre polynomial of degree n. Another special light field derived from the previous cases concerns the superposition of coherent field and a Gaussian field of real or complex amplitude. (i) For real amplitude field

W(p) =

J27l"~E[P] exp ( - 2:[P])

E[pk]

= (2k -

!)

( . T) _ ,(n + p n, - J7rn!

1)!!E[p]k

(2E[p]T)n

(1 + 2E[p]Tt+~

where (2k -I)!! = 1.3.5...(2k -1) and ,( x) is the factorial function, the parameter h being equal to 3. (ii) For complex amplitude field specified by the real parameter

A such that for A

=

0, the resulting field is the superposition of

III. Direct Detection Processing

coherent with a thermal already studied. For A

=

135

1, the result-

ing field is the superposition of coherent with a Gaussian with real amplitude just considered. To see this, use the asymptotic expanSIon z ~ 00: Io(z)

where (n)

rv

e~~Sz;. We obtain

= E[p]T(l -

A2 ), and Pn(x) is the n-th Legendre func-

tion of real argument. Here, E[pk]

=

k!E[p]k(l - ,\2)~Pk(~)

which yields VA, h E [2,3]. Details concerning the special functions Io(x), Ln(z),Ma,p(m),,(x), Pn(x) can be found in [8]. For coherent squeezed light (J-l, v, S), the counting probability density has already been calculated by Yuen [36] for pure input coherent signal and by Helstrom [48] for a input coherent signal corrupted by a thermal background in the case of real J-l, v and for a suitable choice of the phase of the coherent input signal (f) =

p(n;t) ==

I-v

VI -

w2

e

-~s l-w

~

c

v-w

( 1- w

I)

2)n 2

w]

n! [ (1 - v) LJ (n-m)! v-w 2

m

[H m ( C W

(III.l.20a)

m=O

In the case of no thermal background, this distribution takes a simpler form using the conventional notations v = w 2 = tanh( r ),

Se = S - sinh 2 (r),c2 = Si:2~(~~), Hn(c) being the Hermite polynomials [8]

136

Introduction to Photon Communication

pen) =

1 e-Se(l+tanh(r)) cosh(r)

(

tanh(r))n

[H n (c)]2

2

n!

(III.1.20b)

from which we derive 2_s-2r Un -

e

2

+

sinh (2r)

(III.l.20c)

2

that is plotted in the next figure for the three types of light we analyzed. In fact, we must take into account that r depends on S. With the notations

I = ~,

we find r

= atanh (

Clearly, another relation is needed to fix the value of S. This is simply done by seeking the value

of

f yielding the minimum of Pe =

1m

f for given

1 - ...;; e-V¥

rv

tp(O; t). Thus, the variances

for different light fields can be compared. This is done in following Fig. III.lb. 0.3r--~-r------r-----r----r-------,

(a)

0.15

OJ

n 10

15

20

25

Fig.III.la. The curves display the photocount probability distribu-

tions for coherent squeezed state (88) with f = 0.8675, coherent state (C8) and thermal state (TS) versus the photon number n. The received number of photons is S = 4.

III. Direct Detection Processing

137

'-'-

'Of

/v

/

/

J'

./

.&,...,/

/'

/ ./

100

.."

/'

./

./

/"

./

./~

~/

/'" J'

uu

~:

/'" ../

/'" ~ ~

~,/

.....

~)

10-1 10-1

100

10 1

Fig.III.lb. The variances of these fields are plotted in Fig. (b) versus the received number of photons S. To plot (88) the needed value of f for each value of S is taken such that the value of Pe is minimum.

It is interest to notice that the curve fm(B) exhibits a minimum for Ie

"-J

g(s;t)

0.8535 at Be

=

"-J

1.5. Now, let us denote

l-L1(t) 1-s 2 Ll(t)'

where we set t>..

and

= Se, tp = (1 -

h(s;t)=>..t

J)S

I+JL1(t) JLf(ij(l-s) l+s Ll(t)

= Sn, L1 (t)

= l~'tlt. The time

interval distributions are therefore given by [69,70]

v (t) - P (n - 1· t) -

w () t =

-,

gP>(O;t) _ h(l) g(O;t) t (1).

h s (0, t) -

(0· t) ,

g(O;t)

2p(n=2;t) E[p]

( )(0·.2 t) (0· t) + ( ) (0·2 t)

(1) °t(2) (O;t) h(l) (0. t) _ 2 °t (O;t) h(l) (0· t) o(O;t) 8, o(O;t) t ,

h(2) (0· t,

(2) 0, (O;t) h(l)

h(2) 8,

o(O;t)

(III.l.20d)

g~l)(O;t)

(0. t) - 2 0,(1) (O;t) h(l) (0· t) o(O;t) 8,

8,

t) +

h(l) t,

h(l) s,

(III.l.20e)

138

Introduction to Photon Communication

where g~l)(O;t)

= 'g(8jt)ls=0, h~')(O;t) =

'h(8;t)ls=0 and

other similar notations for g~l) (0; t) = :~I g (8; t) Is=o, h~') (0; t)

=

:~I h (8; t) Is=o. A typical example of the distributions (III.1.20d) and (III.1.20e) is displayed below in the next figure.

1

o'--_.......

..-..-_.......__.....__

.-a

o

1

normalized time intervals

3

Fig. 111.2. Time intervals distributions for coherent squeezed light: v( t) waiting time; lV( t) life time, versus time intervals normalised to the average of the total number of photons, for S = 1 and

f = 0.875.

The antibunching effect [59,61,69] is clearly observed for t In fact, from (III.1.15f, g) where E[p]

w(O)

~ v(O) --+

(r7 ~ (p) :::}- a~ ~

=

~

1.

(p)), it is seen that

0, from which it results that the

interpretation of ~uch a non-classical effect occuring in the "intensity" measurements, is inappropriate from the theory of compound random point processes.

III.I.3d Applications As a first application of the results of this section, we want to derive w( tn) the probability distribution that t n == L:~='1 Ti

== 'l9 i +1

-

'l9 i ; Vi, 'l9i

=1=

o.

Ti

where

Let us first calculate this distribution

III. Direct Detection' Processing

139

for a Poisson process. We may use the above expressions or derive it more simply knowing that in this case, the

where we set E[p]

1

w(t n )

21r

are i.i.d.. Thus

= p and with w( Tj) = pe- prj • Therefore,

1

00

00

= -1

Ti

tn(u) e-zutndu = - 1 .

21r

-00

(

-00

p . ) n e-zutndu .

P - zu

As a generalisation of the previous results, we investigate the probability distribution of the sum of a finite number of correlated variables.

a

5

10

15

t

(lJs) 30

Fig. 111.3. Probability distribution w( t n ) of the sum of n intervals for n = 7; T c = 5.65,p,s; E[p] = 5.10 5 s- 1 • (a): Thermal light with a Gaussian profile. (b): Thermal light with an exponential profile. (c): Gaussian distribution E[t] = E(p]; (72 = ;[;1\· (d): Laser light (independent intervals), n+1

W

(t n ) = t t n exp( -pt).

140

Introduction to Photon Communication

As before, let 'Ti == 'l9 i+1

-

'l9 i and t n == E~o 'Ti. The typical curves

of the probability distribution of t n are displayed above. As previously seen, the "r.p.p.", being a compound Poisson process, the

{'Ti} are correlated random variables: eij == E['Ti'Tj]

=I E['Ti]E['Tj]

so

that, Ghanging the notations to read w (t n ) = wn(t; 0), the distribution of the t n for n quite high (in Fig. 111.3, n == 7) is not Gaussian, as it should be for i.i.d.. As a matter of fact, let us be given a set ofr.v. {Xk} where (Xk) == and

1],

a;k == a are their average and variance. 2

Let us consider the r.v. taking the values y = E~l ;~ for fixed M and assume that the r.v. {Xk} are i.i.d.. The central limit the-

orem states that the probability density g(y) tends to the normal

vh: exp( -~). The demonstration can be simplified such that, given px(u) = E [e "\.-'1] which leads to

distribution N(O, 1) =

iU

u + 0 (u-6M! 2M 2

U «P ( )

x

vIM

rv

we will have, for M

3

1- -

--+ 00

and

a;k

u 2M 2

)

rv

exp(--)

> J1 > 0 and (Ixk -

3

1]1 )

<

V

with v > 0,

which is the characteristic function of a normal r.v. In the case that M is random but independent of {x k}, it is simple to prove that for the r.v. z = E~l

(z2) == 1]2(M 2) + a 2(M).

Xk,

we have (z) = TJ(M) and

III. Direct Detection Processing

141

Finally, notice that the central limit theorem holds in certain situations even for not independent, not i.i.d. random variables. The quantum analog of the central limit theorem has already been proved [15]. The demonstration is briefly reproduced in subsection 1.2.8. Therefore, in connection with the application of the central limit theorem, it is of interest to study the deviation of w

(t n ) from a Gaussian whose distribution is, of course, reached

[42]. This is shown in the previous figure where several curves w (t n ) are displayed and compared for various types of

for n

--+ 00

light (coherent, thermal) and for different types of the second-order time correlation functions (Gaussian, exponential). The number of added intervals must be kept constant: 1

~

n

~ 00.

A detailed

analysis of w (t n ) versus n has been published [29]. We may briefly comment on the curves of Fig. 111.3.

1. the curves (a) and ( b) deviate significantly from the Gaussian curve (c) 2. the form of the spectral profile is important 3. for independent intervals (d), the curve approaches the Gauss-

ian curve (c) 4. for (a) and ( b) the behavior can be understood as follows: the

added variables can be split in two sets - the first maximum is related to the distribution of the variables for which ~ij =I- E[Ti]E[Tj]. This occurs for short t. - the second maximum occurs around E[t] = E[p] and can be related to the distribution of the variables where

E[Ti]E[Tj].

~ij ==

142

Introduction to Photon Communication

5. for n ---+

00

most of the

Ti

are i.i.d. and the distribution con-

centrates around the later maximum E[t] = E[p] ,

More details are given in [28,29]. The correlation between intervals can indeed be studied from the coefficients ~ij. It is shown below how to calculate them [1]. Let us denote by

Wn

(tlt9 o = 0)

=

W

(t n ) the probability density

distribution, given that the time interval of length t has been detected, subject to the condition that

'19 0

=

0 and n - 1 points

have occured. As shown in (111.1.16), we have a relation between w n (tlt9 o = 0) andp(n;t).

We want to prove a relation between E[ToTnl and p(n; t). To do this, it is convenient to use the Laplace transforms of p (n; t) and tOn

(tlt9 o = 0). Let

1

00

P(n;()

=

p(n;t)exp(-(t)dt

and

1

00

W(n;()

=

w n (tlt9 o = O)exp(-(t)dt

= Et[exp(-(t)]

By expanding in series the function exp (-(t) it is easy to verify that

~P(n;() ~

I

oo = (_I)k Jo tkp(n;t) dt (=0

~ W (n + 1; () 1,=0 =

(_I)k E

[O:?=o Ti)k]

One therefore obtains

P(n;()

= EJ;] {W(n + 1;() -

2W.(n;() + W(n -1;()}

III. Direct Detection Processing

143

Taking the k-th derivatives of both sides leads finally to:

ak

k ,

8(k P(nj()

1<=0 =

L (_I)k-l ~! (k -I + 1) x 1=0

[(k!'+2 {:;k {W(n + For example, for k 1 P(njO) = E[p]

Ij() - 2W(nj()

+ W(n -

Ij()} }

1<=0]

= 0,

roo 10 p(njt)dt

= ~~ ;2 {W(n + Ij() -

= ~ E [(

t;n

Ti

2W(nj() + W(n -lj() }

) 2 2 (n-l) t; 2+ (n-2t; ) 2] = E [ -

Ti

Ti

TO Tn]

111.2 Detection Performance Because we are mainly interested in detection studies, questions concerning estimation of parameters via the r.p.p. observations are not discussed here. We refer the interested reader to [34,41,43,74].

111.2.1 Detection of Point Processes The most general problem consists of deciding which of the M hypotheses {Hi}~l governs the statistics of an observed path {Yf); 0 ~ {) ~ T}. To simplify the presentation, let us consider a

binary detection of an unknown signal in thermal background of constant intensity The constant value of Po in Ho can be understood because the intensity of thermal noise with a very short correlation time (all the fluctuations are time-integrated), is constant. It can be shown that a rigorous solution to this problem requires the use of the causal minimum mean-square error.

144

Introduction to Photon Communication

As demonstrated, the quasi-optimum solution can be obtained by using a Kalman-Bucy filter (see for example [34]). We emphasize that solving this decision problem yields a complete reconstruction of the required signal [19,20,26]. Unfortunately, implementing the decision by numerical computations appears very difficult to perform. However, for the binary decision problem, some results can be obtained for special situations by studying the likelihood ratio test (LRT) criterion [27,34]. For k

= 0,1, we easily obtain for an equal

a priori probability

for both hypotheses

(i) Ho : Ao(19)

= Po

Po ({ 19 i }i=l)

;

= poe-poT

(ii) HI: Al (19) = PS ( 19) + Po ; n

pt{ {19 i }i=l) =

T

II (ps(19 t) + po)e -poT- Jo i

Ps(t?)dt?

i=l

Hence for the LRT:

(III.2.1a)

where

A({t?i}i=d = Eps

[n (1 + PSp~t?i))

exp

(-iT

ps (t?)dt?) ] (III.2.1b)

Notice that here n is a random variable. The expression of the LRT is expanded

III. Direct Detection Processing

A({19ili=l)

+

:0 t

+-\Po

Eps

.t

rv

Eps

[ex

[ps(19 i )eXp

p

(

(-1

-1

T

PS(19)d19)]

T

E ps [ps (19 i ) PS (19 j) exp

z,}=l

145

Ps(19)d19)]

(-1

(III.2.1c)

T

Ps (19) d'l9)]

0

+... Only the second term of (III.2.1c) depends on the observations. In the case of high level noise po ~ E[ps], n being a random variable, the equation (III.2.1c) can be approximated by the first order term Moreover, it can be shown that for a large class of stationary in -.1... Po processes, this test is a linear function of the observations {{) i} i=l . In fact, for a deterministic signal

(III.2.1d)

the decision is based on the function d ('19) = log ( 1 + pSp~iJ») which can be more precisely studied for small and large signal-to-noise ratios (SNR).

(i) SNR ~1: d ('19)

rv

pSp~iJ) , the solution to the decision problem is

a matched filter receiver of constant impulse response in [0, T]. (ii) SNR ~1: the density Ps (19 i ) can be considered as a random

variable so that Ps ('I9 i ) = niPs,i. Hence L = E~=l ndog P:~i

.

146

Introduction to Photon Communication

Therefore {ni} is a sufficient statistics. The soltltion to this decision problem is the photon counting receiver. It is analyzed in detail in the next section.

111.2.2 Photoncounting Processing : Detection of Coherent Signal in Thermal Noise The detection of a coherent sigl1al in the presence of thermal noise of constant intensity has been extensively studied by Helstrom [37]. Here, we will just mention that closed analytical results for ROC can be obtained for thermal light of arbitrary time correlation~ First for long coherence time and from (111.1.19), the LRT can be proved to verify

A('x)

= exp[-S(1- vt)]L A (-. S(1;t

{ A('x + 1)

1 = 2AH::<1 :;t>2 A('x) -

lld2

)

(III.2.2a)

'xA('x -1)

where L n (x) is the Laguerre polynomial of degree n. We have

A (1)

= (1 + S(1;tlld2 ) exp (-S (1 - vd)

{ A(O) < 1

(III.2.2b)

Therefore, Qd (,X + 1) = Qd (,X) + A('x) [Qo (,X + 1) - Qo (,X) ]. Now for H o and from (11.1.16), Qo (A cause Vt

+ 1) = VtQo (A).

Thus, be-

< 1 we have A(A + 1) > A(A) > O.

Remark: We notice that the curves ROC, i.e. Qq versus Qo are non- decreasing functions and can be constructed segment by segment without any other computation. For an arbitrary coherence time, general results can be obtaiIled from numerical computations using the method recalled below.

III. Direct Detection Processing

147

The light field £ (19) that is considered complex and Gaussian, can be expanded in an orthogonal basis on the interval T

£ (iJ) =

Em CmtP m (iJ)

where

tP m

= (- ~, ~ ):

is a complete basis of the or-

thonormalized eigenfunctions associated with Am, the eigenvalues of the integral equation of the Freedholm type (III.2.3a) with IT ~m (19) ~~ (19) d19

= 6mn .

The statistical properties of the

field allow us to write

The integrated intensity over T is p (T)

=

IT £ (19) £* (19)

E j ICj 2 and its generating function can be written in the form 1

Gt (8; T) = E

[e-SP(T)]

=

L

e -s

2

Ej lej 1 P (C m ) dC m (III.2.3b)

1

=

11 1 + SE[p]A m

m

To calculate the eigenvalues {Am}, the solution to the integral equation (III.2.3a) is required. This can be achieved for the 1£ (19; 19')

= exp (-rl19 -

19'1) typ-

ical second order time correlation function, although in the case where the time correlation function of the light field differs from the exponential profile, some results can be obtained using the numerical method developed by Lachs [25]. The eigenfunctions verify the differential equation

148

Introduction to Photon Communication

We obtain

(III.2.3c) where Xm are the roots of one of the following equations 2xtanx = rT { 2xcotx = -rT

(III.2.3d)

so that only the positive roots are taken into account. Therefore, it is easy to calculate the needed distributions [13]

p (n; T) =

2~ (-It+ k+1 (n) k!p(k; T) D(n-k) (8)1 n! ~ k 8=1 k=O

(III.2.3e) where the following notations are set

I, (() are the I-order modified spherical Bessel functions of the first kind [8], and

Gt

(

8

exp(rT) ; T) = cosh (() + l' ( rT, () sinh (() rT

where we set (= T..jr 2 +2F8E[p] and l'(rT,() = 2(

(

+ 2FT'

On this light field is superimposed a coherent field of constant

III. Direct Detection Processing

149

intensity Pc so that the total generating function is G p (8; T)

=

G t (8; T) exp( -88) where 8 = PeT. Now with (111.1.15e), we obtain

(111.2.3f)

These distributions are usually characterized by the E[N k ]

=

Ilk

moments of N. In some situations, however, other coefficients are more significant. Such coefficients are briefly defined below and some relations with other moments are established. Let cPP ( u) be the characteristic function expanded as follows

(>0

¢>p (u)

= 10

00

exp (iup) W (p) dp =

o

L E[l]

(.

)k

z~!

k=O

= exp [~(iU)k]

def

~Ck-----

k=l

k!

It is simply related to the probability generating function

GN(Z; T) =

~ znp(n; T) =
= G p (l-

z)) = exp

[~Ck (z ~!l)k]

z;T)

Generalized expressions of the characteristic functional and the probability generating functional for a process characterized by an integrated intensity (see Sections 111.1.2- III.1.3a- for 9

I (t; T)

= IT 9 (t -

=

1)

u) dN( u) are studied in [42]. Generalized cumu-

lants can also be derived. From these equations, the integrated intensity moments E[pk], the number moments E[n k], the factorial moments fk, the cumulants

150 Ck,

Introduction to Photon Communication

and the factorial cumulants

9k

are easily derived.

Now, because of the form of ¢p (u), the first-order derivative is sometimes more appropriate to work with


= exp

[

(')k]

LC~ z~! k=l

where

dk

k

dk

c~=dklog¢~(-iv)1 =L~'dk(exP(v)-l)jl Cj v v=O . J. v v=O 1=1

k

(_l)j-m

j

=L .

L m.'eJ _m.),m

k Cj

)=1 m=l

Let us illustrate this by an example in which a relation between

9k

and fk is shown. It is the thermal light with a Lorentzian spectrum [9,28,29], the kth factorial moment is given by

Now, let tJ! (s; T)

tP (s; T)

=-

= log G p (s; T).

~ log (1 + sE[P).\m) = ~ (_l)k :~ E[p)k ( ~.\~ ) k

00

=

From (III.2.3b)

"L..k=l

(-1) k

gk sk!

III. Direct Detection Processing

so that 9k

=

(Em A~)

(k - I)! fk

151

where Am are deduced from

(III.2.3c,d). An interesting expression of 9k that is derived from a recurrence relation

[44] T

L A~ = [~2 dt "'(k) (t, t) m

2

where T

"'(k) (t,t')

= [~2 dxexp(-rlt-xl)"'(k-l)(x,t') 2

Generalization to a M -time joint distribution is now well known. Only some results of calculations for M

=

2, which is the most use-

ful situation for correlation measurements of thermal light source, are recalled here [18]. They can be derived from

where the following notations are set for i = 1,2 [30,50,63] D (Zl , Z2 j T 1, T 2j T)

=[cosh ( Zl T1 ) + (31 sinh (Z2 T 2) ] X [ cosh (z 2T2) + (32 sinh (Z2 T2) ]

~

2 81 8 2

sinh (zl T1) sinh (z2T2) e- 2rr - r (Tl +T2)

J(1+2~81)(1+2~82) where

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Introduction to Photon Communication

For TTl ,2

~

1

9 p (Sl, s2; T l , T2;r) = 1 + E[p] (slTl + +S2T2)

+ E[p]2Tl T2 s l S2 (2 -

II (r))

which yield [12]

L L

(_lfl+T2 A T1T2 P(nl -

Tl,

Tl ; n2 - T2, T2;r) = 0

Tl=O,l r2=O,l

The matrix A being given by

A= (AA A10) OO

Ol

All

(nl) (1 + (n 2 ))) (nl) (n2)

with the boundary values:

The average photon numbers being (ni) = E[p]Ti, i=I,2. A special distribution is obtained for nl = 1; Tl = T2 = T; n2 = n, n

p(l,T;n,T;T)

= Br + B 2

L Br-mB~

(III.2.4a)

m=l

with

B l = A Ol ; B 2 = _1_ (AOl _ All) ; B 3 = A Ol A oo A oo A oo A Ol A oo

(III.2.4b)

III. Direct Detection Processing

153

The previous expression (III.2.4a) should not be compared with the r(n, t == T; T), given by (III.l.17d). The two distributions are different because eq. (III.l.17d) is restricted to T1

---+

o.

This is

the reason why we suggested in [28] to employ the latter method because of its simplicity when one deals with correlation measurements. For higher order time correlation measurements, a generalized time interval distribution can be calculated. It is denoted w (t 1 , ••. tM, ti == 0, Til n == 1) where Ti are the time intervals between the instants time of the point process Ti == 'l9 i+l - 'l9 i. In [47] the calculation of w (t 1 , t2, ti == 0, Tl, T21n == 1) is given in detail. Here, we just recall a few results based on the utilisation of the derivatives of G p (SI' S2; T1 , T2 ; T). For instance, from

it is simple to prove

where we set A( T) == 2 -/I( T) == 1 - exp( -2FT). For the special example M == 2, E[p] ==

r, t1 ---+ 0, rt

~

1,we

obtain w(O,titi

= O,OiTln = 1) =

2E[p] [( ) 2 - E[p]t] 3 1 + 1'1(r) -1 E[ ] (1 + E [p] t ) 1+ Pt

Several probability distributions of interest for various parameters like the bandwidth, signal-to-noise ratio, ... are given in [47].

154

Introduction to Photon Communication

Now these results can be applied to detection purposes. As already noticed, the photon counting processing is the simplest (and often very efficient) method for the applications for communications and spectroscopy as well. The calculations of detection performance are illustrated by plotting two typical figures. They are the ROC and Per (S) curves [14], in the case of the detection of coherent signal (average number of photons S) in a thermal background (average number of photons

No) [48]. The thermal light is of an exponential second-order time correlation function of arbitrary correlation time

Te.

These detection criteria have been defined and analyzed in Sect. II. The needed equations for computing Qo, Qd, Per for binary testing, are given in subsection (11.1.1b) combined with eq. (111.2.3f).

1.0

......,...~-T-----r-----r----r---~

r=O

r=3

Qo 0.0 ' - - - - - - - - - - - - - - - - - - - - - - - ' 1.0 0.0

Fig. 111.4. Receiver Operating Characteristic indexed with the

r signal to noise ratio: processing with relaxed distribution. The average number of noise photons is No = 0.5. For r = 3, rT = 0.1, 1, 10.

From Fig. 111.4, it is observed that the performances improve when r increases. However, for r --+

00,

Qd(Qo)

=1=

1 because of the time

III. Direct Detection Processing

155

integration due to finite coh·erence time of the field. Moreover, we notice on the next figure that the performances improve when processing with q (n) instead of p(n) for several values of S.

0.5 .-----..-------y----y----"T--.------..-----,-----, Per probability of error

q(n) 0.0 L-----1...-_---l....--_.L.-.:.------L----=~~~~~ 0.0 average number of signal photons S 15

Fig.III.5. Minimum probability of error versus average signal photon number. The curve quoted p(n) is obtained with relaxed distribution processing, q(n) with triggered distribution processing. rT = 2.0; No = 0.5

111.2.3 Time Intervals Processing Instead of measuring the number of photons in a given time

[0, T] as done in the previous subsection, it might be useful in some cases to measure the time interval between photoelectrons within the same time duration [0, T]. To show how to calculate the detection performance, let us consider a very simple application detection of a signal in noise, both being assumed coherent. Let t E [0, T] be the r.v. of the nth occurence time of a Poisson process (i) Ho: Poisson process of parameter Ao; Po

=

VQ

(t In)

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Introduction to Photon Communication

(ii) HI: Poisson process of parameter Al VI

(tin) HO

The LRT is equivalent to: t ~ Jln [60]. Hl

Hence Qi For T

= Jci

n

Vi

(tin) dt

(i

= 0,1)·

~ J.l

(111.2.4) where r

= ~:. Here again and as seen in Figs. IV.3-4, Qd increases

when r increases.

111.3 InforIllation PerforIllance In the following, we only consider discrete memoryless optical channels. The input is a random variable X taking M 0,1, ... , M with the a priori probability The output can be the number N(T)

00,01, ... , OM,

=

+ 1 values

respectively.

nl,n2, ... ,oo of received

photons within the time interval [0, T]. With the same modulation format at the input we may choose N (T)

= 0,1. The definitions of

different modulation formats have already been given. In the following subsections, some parameters are introduced to characterize the channels from the information point of view: channel capacity, probability of error and cut-off rate are calculated and analyzed for OOK and PPM modulations in several useful cases. However, some technical questions, important for the design of the system such as synchronization, overlapping,... , are not considered. A review of sensitivity and capacity properties of several photon receivers is given in [59,62].

III. Direct Detection Processing

157

111.3.1 Channel Capacity We start with the fundamental result of point process observations already demonstrated (see e.g. [46,52,55,68,74]). The Poisson channel considered is a continuous-time additive noise channel in which the channel output yt == X t + Nt, t E [0, T] is a Poisson-type point process with intensity

''It

== Xt

+ Po

where

X == X (8, Y) is an encoding of the message 8t , t E [0, T] and Po ~ 0 is the channel noise intensity. The encoder intensity is assumed to be a non-negative, causal, nonanticipative, noiseless, instantaneous feedback of the channel output yt. For a function X (8) that does not depend on Y, the channel has no feedback. The information capacity is defined as 1

Ct == sup sup TIT[8, Y] 8

(111.3.1 )

X

where IT is the average mutual information between 8 and Y. For a channel of no constraint on the coding, the information capacity is infinite.

For a peak constraint 0 :::; Xt :::; Q, it is shown po ( Q ) 1+ -1l Ct = ~ 1 + Po - Po

( 1 + Po ) Q

( Q) log 1 + Po

(111.3.2)

This result has been reconsidered by Davis [52] who included the average constraint E[Xt] :::; Q', and more recently by Frey [75] who studied the problem of mean-square constraint E[X;] :::; Q2. It is interesting to notice that for these types of constraints, the encoding capacity is equal to the information capacity [68]. Different capacity formulas are therefore obtained, particularly the values of

158

Introduction to Photon Communication

the capacity of the noiseless channel Po = O. However they all conclude that, if there is no restriction on X, the information channel capacity is infinite. Now, let us focus on some special channels characterized by the type of signal modulations. 111.3.1a OOK-Modulation

(1) X-channel

1(0) = (1- a)1l(p) + a1l(q) -1l(lP 2 )

(III.3.3a)

For q = 1- p, the channel is called symmetrical. The maximisation of I ( 0) is therefore obtained for a ==

! and the channel capacity

reads

C

= log (2) + H (q)

(III.3.3b)

For the channel, as depicted below, which is a binary photoncount-

ing with soft decision (0 ::; k < 00), the calculations are straightforward.

e {j}

q( klj)

l-aO

Po(k)

a

{k} :0

:1

1 '00

When the source input X

ej

= j, is given with an a priori probability

where j = 1, ... , r, the transition probability distribution for

III. Direct Detection Processing

159

a soft transmission is Pj (k) where k = 0,1, ... ,00. The mutual information between X and N is given by

= (1 -

I (a, X; N)

a) ?to

+ a?tl -?tOl

where Hi = E~=o Pi (n) log Pi (n) with i=O,l and HOl = E~=o IP (n ) log IP (n) {

IP ( n)

= (1 -

The channel capacity will be

a) Po (n)

cf =

+ aPl

(n)

max I(a,X; N). Now, suppose a

one restricts the received photon number N (T) = n to be binary

{t; n = 1 for n > {t. The mutual information can be written in the form

outputs such as n = 0 for n

~

with the following notations JC (1r ({t, a))

= -1r ({t, a) log (1r ({t, a))

- (1 - ?T ({t, a)) log (1 ?T ({t, a) = (1 - a) So ({t) + aTt ({t) So ({t) = E~=o Po (n) Tt ({t) = E~=o Pt (n) To ({t) = 1 - So ({t) The capacity will be

cf

?T

({t, a))

= maxa,1L1(a,{t) = max lL C*({t). The

maximum over a exists and is reached for •

a (J.L) where

So (J.L) [1 + exp (X (J.L ))] - 1

=--------[1 + exp (X (J.L )) ] [So (J.L) - Tl (J.L) ]

160

Introduction to Photon Communication

is a decreasing function from X (0) to -

00.

A graphical study

of C* (fl) is convenient.

JC(E) 0.7

1.0

Fig. 111.6. Graphical representation of JC (E) and C* (p,) == 12 (p,) 11 (p,).

With the help of Fig. 111.6, it is shown that C* (fl) == 12 (fl) - 11 (fl)

== log [1 + exp (X (fl))] - X (fl) + X (fl) So (fl) - JC (So (fl)) Hence the maximum is attained for

with X (fl) ~

o.

This is approximately verified (\if p" T1 (p,) which JC (T1 (flo)) == JC (So (flo)).

~

So (p,)) for fl * ::; flo for

III. Direct Detection Processing

161

(2) Z-channel The a priori probability vector can be written as Q == (a, I-a)t.

The P (YIX) =

(1 ~

p

~) is the probability transition matrix and

the mutual information is given by Iz (a; X, Y) == a1i (p) -1i (ap) where H(p) == plog(p)

+ (1- p)log(I- p)

is the binary entropy

function. To calculate the capacity, we set the derivative of the mutual information versus a equal to 0:

{) oa Iz (ai X ,Y)

=0

~ 1i(p)

)P = log ( 1-a*p a*p

Therefore

Cz(p) = Iz (a*iX, Y) = log

(1 + p(l- p)7)

Let us consider the Z-channel for which Po (n) == 8 (n). A closed form for

cf can then easily be derived. It is shown that only one

maximum exists, obtained as before by setting the derivative with respect to a equal to zero. · occurs at a = a * =. ThIS

1 (Pl(O)Pl(O») I-pl(O)+exp l-Pl 0)

Pl(O)---+O

) -1 e

W h·lC h

leads to

(III.3.3c)

(3) Comparison It is interesting to compare from the capacity point of view several types of channel that we already mentioned. We consider: (0) Holevo bound (i) photoncounting channel,

162

Introduction to Photon Communication

(ii) quasi-classical channel, (iii) quantum channel, (iv) phase channel. The Holevo bound is derived from Eq.(11.3.2) with ~i == ~. The eigenvalues of

l7Po) (7Pll + l7Pl) (7Pll

are calculated as in Sect. (11.2.1)

so that

cb = where, ==

log 2 -

~ ((1 + 'Y ) log( 1 + 'Y) + (1 -

(7Po l7Pl) == e- t

'Y) log( 1 - 'Y))

for pure real coherent states. For the last

three models, the channel is a symmetrical one and characterized by a probability transition respectively given by

(III.3.3d, e, f)

p(¢) being given by eqs.(IV.13a,b). For the case (i), the channel is of Z-type and the capacity was calculated in Eq. (1II.3.3c). For the other cases, the capacity in nats, is given by C == log(2)

+ Pi log(pi) + (1 -

Pi) log(l - Pi),

(i == c, q, ¢). (111.3.3g)

The corresponding curves versus the average photon number S ==

Ifll 2

are displayed in next figure. Notice that for (i)-(iv), the mod-

ulation is PSK.

III. Direct Detection Processing O.?

163

nats

0.6

0.5

0.4

0.3

3:

cc

4:

C4>

0.2

O.l

average number of photons 10- 1

S

100

10 1

Fig. 111.7. Capacity versus S for various channels.

It is clearly seen that the receiver based on optimum quantum detection is the closest to the upper bound. It is superior to the counter, the .homodyne receiver and the phase receiver.

III.3.1b PPM-Modulation (1) Soft Decision Let x E Ax = (0,1) and y E Ay = lN M = {ni}f!l. channel capacity can be written, with (i C 8 --

~ ~ ~P ro b( y Ix )1 og

L-J L-J M xEAx YEAy

L

= k,

Vi

Prob(ylx) 1 M Prob (yle)

eEAx

With the independence assumption

p(yIXi)

= IIpo (nj)Pl (ni), j=l j¢i

The

Vi

(III.3.4a)

164

Introduction to Photon Communication

it is shown

{ni~l=O {Pdnt) gpo (nj) log (t ~~::~) }

C s = logM -

(III.3.4b)

where A(nk) is the likelihood ratio (see 111.1.1). For a Z-channel

C s = [1- PI (O)]logM

(111.3.4c)

(2) Hard Decision The channel is defined by the following transition probabilities

Vi

1, ... ,M

=

(i) i = j, (ii) i

=1=

j,

pdef

Prob(Y = i\X = j)

/ef Prob (Y = ilX = j)

(iii) Vj,j E lN M , f

def

Prob(Y = fiX =j)

(iv) p+(M-1)q+€=1

p=

~Pdn) [~PO (k)]

q=

~PO (n) { [~PO (k)]

M-l

M-2

~Pl (k)}

Therefore, it is easily proved

Ch = (1- €)logM - (1- €)log(l- €) + plogp+ (M -l)qlogq It is demonstrated in [67] that M for q

= 0, p = 1 -

€

= 1-

~

1,Ch

~

plogM. Notice that

PI (0), we obtain Ch

= Cs.

III. Direct Detection Processing

165

(3) Two Results It is important to notice the following results: 1.

Poisson distributions:

For a weak noise channel, with an average power constraint, and a bandwidth constraint, it is proved that there exists M*, the optimum alphabet size of M, such that 2(1+ 2N o)

2

8 - 8 2.

-4

0, MNo=o

-4 00,

No

~

rv

e s , MNo~1

1, M*

rv

rv

e

S

3 [66].

Coherent signal in a thermal noise of long coherence time:

This example was extensively studied by Pierce et al [45,53]. It has been proved that the Cs soft decoding channel capacity is bounded

by

C~

= log ( 1 +

Jo) for an average power constraint.

This result can be generalized to an arbitrary coherence time for an exponential second order time correlation function by using the Chernov bounds method [67]. Let us briefly explain how to obtain such generalization. Denoting Gi(Z) = L:~oPi(k)zk, (i = 0,1) the generating functions associated with the distributions Pi (k) for a binary channel with

Izl ::;

ei and where ei ~ 1 are the abscissa of convergence of

Gi(Z). For a wide class of light fields, we have

{

G1(Z) = Go(z) exp (-8g (z)) g(l) - 0 dg(z) I dz z=1-

-1

where Po(k) is the distribution characterized by the average photon number No and PI (k) is the distribution characterized by the average photon number N 1

= No

+ S.

166

Introduction to Photon Communication

For example, Go(z)

= L:~o(1- Vt)vtz k = l~~'z with Vt = 1fNo

for a thermal light and

=

z ) Go(z)exp ( -S(l- V t1)--

1-

VtZ

where z E [1, ~t [ for a superposition of a coherent light and an independent thermal light. The Chernov's bound [7] reads here 00

00

LPo(k) ~ L zn-kpo(k) ~ z-nGo(z) k=n

k=n

Now, recall the formula established for the probability of error in the case of P·PM-modulation (11.1.4)

Per

~

min

6E[O,1]

zE[l,eo]

~ P1(1) [1 _(1- ~ po(k)) M-l] LJ LJ 1=0

k=1

~ ~P1(1) (~po(k)) M6

6

where we used

1- (1- A)M-l < M DAD, '18 E [0,1], VA E [0,1] Moreover, it is simple to prove as long as the rate def

RM =

logM

S

I

t

~ og<"o

III. Direct Detection Processing

an optimum value M* of M exists for which Per

---t

o.

167

Therefore

the maximum value of RM is indeed the channel capacity. So, we may state

C = log eo

- log

1

Vt

= log

(No) 1 + No

Let us apply this theory to the case of the superposition of the coherent light and the thermal light of exponential time autocorrelation function ,(T)

J"

= exp(-TITI) with arbitrary coherence time

the corresponding abscissa of convergence are shown to verify

(see III.2.3d)

eo = 1 + max· 1(( ) . I

with W =

r[,

I

and where T is the time duration of measurement

and No the average value of photon number of the thermal component. ~

1,

rT~

1,

1. rT

2.

Xi

rv

W

From the above discussion, we conclude that

Cs

:s; log

( 1+

showing, for finite No

TT

---t 00

rT

---t

0

rT) N---:-

1+

Poisson channel Pierce result

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Introduction to Photon Communication

111.3.2 Cut-Off Rate We briefly recall the coding theorem [6,7,40] for a discrete memoryless channel without cost constraint. For each pair Xl, X2 E Ax, we consider

J(Xx,xz) =

l:yEA y

vp(ylxt)p(ylxz)

J o == min{E[J (Xl, X2)]} Q(x)

R o == log ( J0 ) For a binary channel where Ax == (0,1), p(yIO)

= Po (n),

p(yI1) -

PI (n), the following expressions are obtained.

(1) Soft decision As is shown hereafter, this cut-off is interesting because it is a limit for the hard decision processing. The calculation is straightforward 2 R os == Iog---oo:-:----====== 1 + l:n=O (n)PI (n)

vpo

(2) Hard decision The needed distributions are all already given in previous subsection. Here again, the calculations are very simple h 2 R o == log---;::::=====---;::===== 1 + VSo (~) TI (~) + VSI (~) To (~)

(3) Some general results The hard decision performances are always worse than those obtained from the soft decision. To prove this, it is sufficient to apply the Schwarz inequality:

III. Direct Detection Processing 00

J~

=L

Jpo(n)P

1

I.t

I.t

n=O

n=O

169

L Po (n) L P (n)

(n) ~

1

n=O

00

L

+

00

Po (n)

n=I.t+1

L

Pd n)

n=I.t+1

= J~

Therefore, we can conclude Vj.l, Ro ~ Rg In the M -ary PPM signalling as proposed by Pierce for optical communication [45,53], the time axis is divided into M time slots and the light pulse is only present during the 111,th slot corresponding to the mth symbol. At the detector, synchronized to the PPM modulation, we receive {n }~I photoncounts which are M independent random variables. If the message

mi

is sent, let Po (n) be

the probability that the light pulse is in the jth time slot for j

=1=

i.

Questions concerning time synchronization are extensively studied in [38] (pp. 329-367). Let PI (n) be the probability that the light pulse is in the j th time slot for j

=

i. The expression for the cut-off for a soft decoding

receiver is given by [49,56,57,65]

R o = -log min

L

{Qm} {}M n i=l

= log M -

log

(f Jp({n}~1Im)Qm)2 m=1

(1 + (M - 1) (~J

Po (k) Pl (k))

For Poisson distributions, the cut-off rate is [56]

2)

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Introduction to Photon Communication

R o = log M - log

[1 + (M - 1) exp ( -

(

VS + No _ ~) 2)]

An analysis of this formula was made by Chan [51]. We recall the main results -

With bandwidth constraint -

for No

= 0,

M

---+ 00,

R o ---+ log 2

- for No ~ log(M), R o -

rv

S Vlog(M) No

With average power constraint and finite bandwidth, M*

IS

such that

-

S

~

-

S

rv

1, M*

2 (1

+ J2No)

rv - - - - -

S

1 + V2No, M*

rv

3.

These conclusions are only in qualitative agreement with those derived from the capacity analysis [66]. Finally, this subsection on photoncount processing is closed with an important theorem which is stated and its demonstration just outlined. For the detection of coherent signal in thermal noise of long coherence time, the cut-off rate is bounded.

For S ~ 1, R o rv

4 lit (f+lId

To find the bound for S

·

~

1, we proceed as follows:

- first, notice that R o ~ -Sl min {GO(z)G 1 zE[l, lilt

[

(~)

}

Z

- then,

(~ Vpo (k) PI (k)) ~ (1 - Vt)2 e2

S

(l-lI t )

x

~v;k Lk ( _ B(l ~ Vt)2)

III. Direct Detection Processing

171

- use now the inequality verified by the Laguerre polynomials for

x

~

0, i ~ j, that is Li (-x) ~ Lj (-x), to obtain

(~Vpo (k)Pl (k)) ~ exp ( -1 +~No) 2

- finally, 1
R <

Comparison of the previous results show that the cut-off rate and the capacity points of view are quite different. The capacity criterion appears very difficult to attain. Now, as already mentioned in III.2.2b, the time interval processor is often a preferable mode of operation because of its simplicity. Furthermore, in some situations, it performs better than conventional photon counting processing [72]. To illustrate this situation, let us consider two types of light of different second-order properties:

(i) thermal light exhibiting bunching, (ii) squeezed light exhibiting antibunching and examine the effects on the cut-off rate for these two specific examples. Consider a Z-channel for which a correct detection of a "0" is always possible and the only error comes from the detection of the symbol "1". Assume that the receiver measures the time interval between photoelectrons. The transition probabilities are p(OIO)

8(t); p(111)

= z(t),

the time interval t being such that t

=

< S.

For soft-decoding, for which the fundamental limit is attained, the cut-off rate takes the form

172

Introduction to Photon Communication

RS(S) = log 2 o 1 + ft) dty'S(t)z (8 - t) The first case (i) (Fig. 111.8), corresponds to the source of information is a thermal light of long coherence time, for which

z(t)

= v(t)

given by (111.l.18a) yielding the curve denoted R v ,

(resp. w(t) given by (111.l.18b) yielding R w ). It is clearly seen that for S

~

1, processing with life distribution yields higher per-

formance. This is due to bunching effect which allow for t

~

1

(i.e. S ~ 1), to have w(S) ~ v(S).

0.6

Ro (nats)

0.4

0.2

O.O'----_ _ 0.1 average

----L

----L-

number of signal photons S

----J

10

Fig. 111.8. Thermal signal. Cut-off rate versus S the average number of the received photons. R v : processing with the waiting time distribution. R w : processing with the life time distribution.

Now, in the second case (Fig. 111.9), the information is carried out by squeezed light for which z (t) is v (t) given by (III.l.20d), (resp. w

(t) given by (1II.l.20e)). Because S depends on the degree of

squeezing, this parameter must be optimized (e.g. by minimizing the probability of error).

III. Direct Detection Processing

173

Such a constraint on information criterion has already been considered for different purpose [10]. We observe that VS processing with waiting time distribution leads to higher performance. This is due to the antibunching effect which tends to force the photons apart. More details of the calculations are given in [71].

R

(na~s l.Ou--~-~-----, 0.6 w

0.4 0.8~--~------'

1.0

0.2

0.0 0.1

L.......-

--L-

average number of signal photons S

Fig.III.9. Coherent squeezed

---'

10

light~

Cut-off rate versus S the average number of the received photons. R v : processing with the waiting time distribution with minimization of the probability of error for Iv (5). R w : processing with the life time distribution with minimization of the probability of error for fw (S) .

111.4 References [1] [2] [3] [4] [5]

[6]

J.A. McFadden: Jour. Roy. Soc. B 24 364 (1962). R. Glauber: Phys. Rev., 131 2766 (1963). P.L. Kelley, W.R. Kleiner: Phys. Rev. A, 30 844 (1964). L. Mandel, E. Wolf: Rev. Mod. Phys. 35,231 (1965). A. Papoulis: Probability, Random Variables and Stochastic processes (Mc Graw-Hill, New York 1965). J. Wozencraft, I. Jacobs: Principles of Communication Engineering (John Wiley & Sons, New York 1965).

174

Introduction to Photon Communication

[7]

R.G. Gallager: Information Theory and Reliable Communication (Academic Press, New York 1965). I.S. Gradshteyn, I.M. Ryzhik: Table of Integrals, Series and Products (Academic Press, New York 1965). R. Glauber: Physics of Quantum Electronics, ed. P. Kelly, B. Lax and O. Tannenwald (McGraw-Hill, New York 1966). T.L. Gabriele: I.E.E.E. IT-12 484 (1966). D.R. Cox, P.A.W. Lewis: The statistical analysis of series of events (Methuen, London 1966). F.J. Beutler, O. Leneman: Acta Math. 116 159 (1966). G. Bedard: Phys. Rev. 161 1304 (1967). H.L. van Trees: Detection, Estimation and Modulation Theory (John Wiley & Sons, New York 1968). Part I. J.R. Klauder, E.C.G. Sudarshan: Fundamental of Quantum Optics (Benjamin, New York 1968). B.R. Mollow: Phys. Rev., 168 1896 (1968). F.T. Arecchi: Photocount distributions and field statistics, (International School of Physics" Enrico Fermi", Varenna 1968). D.B. Searl: Phys. Rev., 175 1661 (1968). T.E. Duncan: Inf. Contra 13 62 (1968). T. Kailath: I.E.E.E. IT-15 350 (1969). B. Picinbono, M. Rousseau: Phys. Rev. A, 1 630 (1970). B. Picinbono, C. Bendjaballah, J. Pouget: Jour. Math. Phys. 11 2166 (1970). J. Perina: Coherence of Light (van Nostrand Reinhold, London 1971). E.B. Davies: Comma Math. Phys. 22 5 (1971). G. Lachs: Jour. Appl. Phys. 42 602 (1971). I. Rubin: I.E.E.E. IT-18 547 (1972). O. Macchi: Processus ponctuels et coincidences, These de l' Universite d'Orsay (1972). C. Bendjaballah: Analyse de champs optiques par les methodes de comptage et de coincidences de photons, These de l' Universite d'Orsay (1972). C. Bendjaballah, F. Perrot: Jour. Appl. Phys. 44 5130 (1973). S.K. Srinivasan, S. Sukanam, E.C.G. Sudarshan: Jour. Phys. A 6 1910 (1973). R. Loudon: The Quantum Theory of Light (Clarendon Press, London 1973). E.V. Hoversten, D.L. Snyder, R.O. Harger, K. Kurimoto: I.E.E.E. COM-22 17 (1974).

[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

[29] [30] [31] [32]

III. Direct Detection Processing [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63]

175

R.O. Harger: I.E.E.E. AES-ll 629 (1975). D.B. Snyder: Random Point Processes (Wiley, New York 1975). H.P. Yuen, J.H. Shapiro: I.E.E.E. IT-21 125 (1975). H.P. Yuen: Phys. Rev. A, 13 2226 (1976). C.W. Helstrom: Quantum Detection and Estimation Theory (Academic Press, New York 1976). R.M. Gagliardi, S. Karp: Optical Communications (Wiley, New York 1976). R.O. Harger: Optical Communication Theory (Dowden, Hutchinson & Ross, Stroudsburg, (Pennsylvania). 1977). R.J. McEliece: The Theory of Information and Coding (AddisonWesley Pub. Co., Reading (USA). 1977). R.S. Lipster, A.N. Shirayev: Statistics of Point Processes II. Applications (Springer Verlag, New York 1977). J. Rice: Adv. Appl. Probe 9 553 (1977). P. Bremaud, J. Jacod: Adv. Appl. Probe COM-9 362 (1977). B.E.A. Saleh: Photoelectron Statistics (Springer-Verlag, Berlin 1978). J.R. Pierce: I.E.E.E. COM-26 1819 (1978). Y. Kabanov: Theory of Probe & Appl. IT-23 490 (1978). C. Bendjaballah: Jour. Appl. Phys. 50 62 (1978). C.W. Helstrom: I.E.E.E. IT-25 490 (1980). D.L Snyder, I.B. Rhodes: I.E.E.E. IT-26 327 (1980). R. Barakat, J. Blake: Physics Rep. 60 225 (1980). V.W.S. Chan: I.E.E.E. NTC-3 57.4.1.-57.4.8. (1980). M. Davis: I.E.E.E. IT-26 710 (1980). J.R. Pierce, E.C. Posner, E.R. Rodemich: I.E.E.E. IT-27 61 (1981 ). M.D. Srinivas, E.B. Davies: Optica Acta 28 981 (1981). P. Bremaud: Point Processes and Queues (Springer-Verlag, Berlin 1981). R.J. McEliece: I.E.E.E. IT-27 393 (1981). J.L. Massey: I.E.E.E. COM-29 1615 (1981). A.S. Holevo: Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam 1982). H. Paul: Review of Mod. Phys. 54 1061 (1982). C. Bendjaballah, K. Hassan: Jour. Opt. Soc. Amer. 73 1840 (1983). B. Yurke: Phys. Rev. A, 32 311 (1985). Y. Yamamoto, H.A. Haus: Rev. of Mod. Phys. 58 1001 (1986). M. Singh: Optica Acta 33 855 (1986).

176

Introduction to Photon Communication

[64] [65] [66] [67] [68] [69] [70]

M. Cini, M. Serva : Jour. of Phys. A: Math. Gen. 19 1163 (1986). M. Charbit, C. Bendjaballah: I.E.E'.E. COM-34 600 (1986). M. Charbit, C. Bendjaballah: Optics & Quant. Elect. 1849 (1986). M. Charbit, C. Bendjaballah: I.E.E.E. COM-35 122 (1987). A.D. Wyner: I.E.E.E. IT-34 1449 (1988). R. Vyas, S. Sing: Phys. Rev. A, 38 2423 (1988). D.J. Daley, D. Vere-Jones: An Introduction to the Theory of Point Processes (Springer-Verlag, Berlin 1988). C. Bendjaballah, H. Ie Pas de Secheval: Optics Comm. 74 403 (1990). H. Ie Pas de Secheval, C. Bendjaballah: Coherence and Quantum Optics VI, ed. by L. Mandel and E. Wolf (Plenum Press, New York 1990). p. 675. A. Barchielli: Quantum Aspects of Optical Communications, ed. by C. Bendjaballah, O. Hirota and S. Reynaud (Springer Verlag, Berlin 1991). A.O. Hero: Proc. of the Conf. on Information Sciences and Systems (John Hopkins, Baltimore, USA 1991). M.R. Frey: I.E.E.E. IT-37 244 (1991).

[71] [72]

[73]

[74] [75]

IV. Phase Operator

The problem of a phase operator, which obeys the laws of quantum mechanics, has inspired many physicists since Dirac first assumed that a Hermitian phase operator, canonically conjugate to the number operator, existed. On the other hand, the study of squeezed states of light has renewed interest in quantum phase operator measurement. Numerous studies have been devoted to this subject but several questions remain unsolved. It seems that no Hermitian operator corresponds to the phase of a quantum electromagnetic field. One of the problems one can examine, is to find which phase probability distribution is relevant to the observation of quantum phase. We nlay also wonder what the quantum phase operator, analyzed in [1], stands for, how it is specifically related to the quantum phase observable introduced in [2] and extensively discussed in [5]. Another question relevant to the wave packet reduction postulate, is to describe the state of the quantum oscillator after having measured its phase. All these questions are obviously essential from a theoretical point of view. However, they deserve· to be studied from a practical point of view too. Indeed, in order to apply techniques such as heterodyning, to use squeezed light or phase shift keying modulation in optical experiments involving phase measurement, it is important to answer these questions and to determine the quantum limit performance for optical communications based for example on phase measurement. Numerous authors have already proposed solutions which, however, differ significantly in many points, e.g. phase probability dis-

178

Introduction to Photon Communication

tribution and the commutator of photon number-phase operators. The problem comes essentially from the difficulty of defining an appropriate quantum phase operator dynamically conjugated to the photon number due to the fact that the shift n

n

1---+

n

+ 1 of

= 0,1,2, ... cannot be inverted and hence does not generate a

unitary group in the corresponding Hilbert space.

IV.1 SOIne Models The first method proposed by Susskind and Glogower (see

[1])

= 8 + is as the phase operator via the sine and cosine operators s,8 such that the (non-normalizable) phase states are defines fj

given by 00

Iw(cP)) = lexp(icP)) = LeintPln)

(IV.I)

n=O

defined in terms of the number states In). This model provides a satisfactory description in the region of high energy states and leads to the p.o.m.

iiJ1 (cP) = 2~ with Ll

=

[,

i

Iexp(icP)) (exp(icP)ldcP

(IV.2)

+ 21r).

However, because the cosine and sine operators do not commute

[8, S]

= -lIO)(OI =1= 0, it is not possible to find a complete set of

eigenvectors of 8 and is only approximate.

Ssimultaneously. Therefore this description

IV. Phase Operator

179

In favor of this model, a remark made by [3] that 00

L In)(n + 11

E=

(IV.3)

n=O

with eigenstate I¢»

=

1 2 1r

L:~=o e in 4>ln) is the appropriate operator

=I Et), if E could be adopted for the same reason that an observable a is accepted despite that a =I at. although it is not hermitian (E

Another approach is based on the maximum likelihood ratio criterion of the quantum estimation theory [6,10], to obtain the p.o.m. and the conditional probability p( ¢>16) of measllring the phase estimate ¢> given that the true value is 6

dfi(
= 21

1r

p(¢>16)d¢>

where Nlk)

=

exp( -i~)1-)') h'1 exp( iN )d

(IV.4)

= (~(6)ldlI(¢»I~(6))

klk) and 11) is defined in terms of projections of

states of the system in 1~(6))

= L:~=o ~n exp(in6)ln)

tion of the unknown parameter 6 (p(6)

=

a state func-

1~(6))(~(6)1),

into Ik).

To demonstrate (IV.4), some elements of the quantum estimation theory are recalled in Sect. 11.2.1. (see eqs. (11.2.3a-e)). Obviously, this description strongly depends upon a special criterion of quantum estimation theory which implies that the p.o.m. is not optimal in either strategy, for instance Shannon's mutual information [19]. Nevertheless, to prove that this model is useful, we study in which conditions the operators Nand

according to Kraus [4]. Two observables

X,

¥ are conjugate

and jj of eigenvectors

{Iai)} and {lb j )} in a Hilbert space of dimension d are called complementary if 'v'i,jl(ailbj) = ~.

180

Introduction to Photon Communication

Furthermore, the relations between optimal detection operator and phase shift measurement are analyzed [7]. Let IJ-lk)

-

IJ-lu k)

=

IJ-lexp (_i2~k)) be phase modulated

(mod. d) coherent states given with probability a priori {(k}d-l k=O

=

where u

exp (-i

shift operator IJ-lk) 9ij

= (J-lilJ-lj) (1 -

exp[-J-l2

2;) =

p

~

d - 1 and 0

~

k

~

pk ... u-P(d-l)]

(1 -

2;),

the

, ,

+ fliflj) =

t

(IV.5)

d - 1.

The eigenvalues are given by Ap exp [-1l 2

exp ( -ikN

is hermitian and circulant 9ij = exp (_fl2 u j - i )]. From GI,8p) = Ap l,8p), we obtain

[1

~

=

WklJ-l). The Gram matrix G of elements

1,8p ) = ~ -Id' u- P, ... ,ufor 0

~

and denote

"d-l h Lik=O 90k U - pk were 90k

uk)].

We now define X so that

xt X =

G, and using the spectral

decomposition theorem, we set d-l

G=

L Apl,Bp)(,Bpl p=o

X =

d-l

d-l

p=o

p=o

L epl,Bp) (,Bpi =L A

exp(i8p)l,Bp)(,Bpl

--- = Iw

Let us examine in which case the p.o.m. fl m

m ) (w m

I,

l:m ~ = n, are the optimum detection operators (0. d. 0.). Given the results of the quantum detection theory (see Sect. II), detection probability Qd is maximum for

IV. Phase Operator

181

(IV.6a, b, c)

where we set

Illk)

X

rs

(,BrIXI,Bs). To evaluate the Ap , we start with

=

= exp ( -ikN

= ~ exp ( -ikr 2;) Ir)(rlll)

2 ; ) Ill)

(IV.7)

and taking into account that

where I:~:~ u kr =

°

if k =F 0, d if k = 0, we arrive at

d-l

Then as

X rs

=

~L

Au-psu

pr

we readily obtain

p=O d-l

(nlwm) = 'Lelm(nlJ-ll) [=0

1 S = -ex p (--) d 2

/fr

d-l d-l

Sn 1 2i1r - " " - e x p [-(k(f-m)-fn)] n! L..J L..J y'Xk d [=0 k=O

(IV.9)

Therefore

l(nlwm)1

= exp

~ A (-"2S) y-:;;;:

1

=

1 [ Sd ] Jd 1- 2(n + d) + ...

(IV.IO)

182

Introduction to Photon Communication

n;;: are com-

This result demonstrates that the observables Nand

plementary for Sd ~ 4d, that is verified for S ~ 1, Vd ~ 1. Let us analyze the asymptotical behavior with respect to d. As SP

sp+nd

L (p + nd)'. 00

r-.J

n=O

-, '

p.

AP

d

=

jj2 p exp( _jj2)_,

= (JLI,Br). If we set I
=

jr;1,Br), Xro

it can be seen that lim d--+oo

p.

which is a Poisson distribution.

From eq.(IV.8), we get Xro can be rewritten as (with c.p

w( c.p)

= 2; r)

d-l (- JL 2) JPfJL exp (-ic.pp)

= (jjl¢» = d--+oo lim y'2; L exp 21r 1

p=O

P

2

p.

(IV.II) where 1

I
L exp (-inc.p) In) 00

tiC

(IV.I2)

V 21r n=O

which is similar to eq.(IV.I). The probability distribution of the phase shift 1 . P(c.p) = Iw( c.p )1 = 271" 2

2

00

IL exp (-inc.p) (JLln) I

(IV.I3a)

n=O

takes the form

Pcs(c.p) =exp(-IJLI 2 )

L p,q=O

1

-,-,JLp+qcos[(p-q)c.p]

(IV.I3b)

p.q.

using (IV.IO) in the case of a coherent state (CS). Remark that, owing to the central limit theorem (see § III.I.3d),

1(/1ln)1 2 can be approximated by a Gaussian form for /1

~ 00, the

IV. Phase Operator

183

distribution (IV.13b) will therefore exhibit a Gaussian behavior. The calculation of trle phase distributions for other models is continued for thermal light (T8) for which 1- y2

1

P

TS

(r.p)

= -21r 1- 2ycosr.p + y2 ,

where (JLln)

=

y

~n

1-

VV2,

n

(IV.13c)

= 0, 1, ... ,00

with v

=

~

1+11t1 2 • For a

space of finite dimension d, we obtain [18] 1 1 - y2 1 - 2y d+l cos( (d + 1)r.p) + y2d+2 (r.p) - - - - - - - - - - - - - - - TS 21r 1 - 2y cos r.p + y2 1 - y2d+2

P

(IV.13c') For pure squeezed light (88), the phase distribution is derived from • (III.1.20b) wIth Be

(JLln) =

2r

e S = IJLI 2 ,c2 = sinh(2~)

1 e-.2t (1+ tan h r) vcoshr

(

tanh r) - 2-

y;]

~

Hn ( c)

(IV.13d)

which is put into (IV.13a) for numerical computation. These three distributions, which are symmetrieal in r.p, will be displayed in Fig. IV.la together with the square root of the product of variances of nand r.p in Fig. IV.lb. To compute the square root of this product in an approximately closed form, we may consider, as in the case of x and p variables (1.2.26), a joint distribution of the conjugated variables. The number of photons n and phase r.p can be obtained from a Wigner distribution (IV.14)

184

Introduction to Photon Communication

which has been evaluated in the case of a highly squeezed state [8].

1.6 1.4

~

\

1.2

._JJ ._~

0.8

(G

{i>) 'L·~t~

,S

\

~ """"

0.6

'\:\

\~\

0.4

\

0.2

o2

10-

\ \\\\

'.\~'10- 1

(1) 10 1

10°

Fig. IV.la. The curves display the phase probability distributions for coherent squeezed state (SS) with f = 0.8555, coherent state (CS) and thermal state (TS) versus the phase ¢. The received number of photons is S = 2.

1.4

1

1.2

0.8

0.6

Un UA.

=1

\

\

\

"

.........

"'--

------ - -- - :----

0.4 0.2

3.5

S 10

12

14

16

18

20

Fig. IV.lb. The product of the square root of the variances O'nO'4> are plotted versus the received number of photons S, O'n being given by III.1.20c for the SSe In this case the curves p(¢), parametrized with S = 0.1,4, 12, undergoe a bifurcation (see [8]).

IV. Phase Operator

185

Computation of (J'n(J'~ derived from eq. (IV.13b), for curve (CS),

shows that

r

(In(J4>

~ ~. To compare with curve (88), we set

= atanh (

and

f

rv

1-

fe-V"il.

approximate calculations show that for S

~

In this case,

1, (J'na¢

rv

0.6 [17].

This asymptotic value is close to the value that one can obtain by extrapolating the values derived from the p( ¢» (see curve SS of Fig. IV.1a). The deviation might be due to the method used to fix

f

and to some lack of precision in numerical computation. Now, if S are r are independent, as it is in Fig IV.1b, then

ana¢

---+

0.5.

The corresponding curve for the (TS) is continuously increasing (in the range of the S considered he~e), due to the very slow decrease of the thermal photoncounting distribution. All the results obtained here seem in a satisfactory agreement with the models (e.g. [9,13,17]), except in the region of a weak S. However, it must be pointed out that the probability distributions are not accurately normalized to 1, because of a lack of precision in the numerical computation. In the third approach of the phase operator, recently proposed in

[5], Pegg and Barnett introduce the phase operator from a finitedimensional space

(IV.15) f) k(d) --

f)

0

k + 21r d+l

186

Introduction to Photon Communication

The state-space dimension d may tend to infinity only after physical results such as expectation values have been calculated. Thus, defining a mapping between any function of operators ~c and states l"p)c of the conventional Fock space 1i, {In)} and those defined in the truncated space 1id [12]

(IV.16) This mathematical construction is interesting because these phase states form a complete orthonormal basis. Furthermore they yield, among others, the following properties exp (i¥d) 10)

= e(d+l)8 Id) 0

q

(nl cos

2q

In)

= (nl sin

2q

In)

=

II

m=l

2m-1 2m

(IV.17)

which are very important and not given by the previous models. As a simple application, let us briefly point out some remarks concerning the entropy of the probability distribution of coherent phase. Instead of utilizing Shannon's entropy 84>( n) for d --+

00,

= log d which diverges

due to the complementarity property (IV.IO), it is

suggested to work with the relative entropy 84>( n) + log q( d) which is bounded for q(d) = ~. The choice of this a priori probability is motivated by the fact that the knowledge of the dimension of the space must be taken into account. On the other hand, it is of interest to evaluate how thermalization of the state changes the entropy. When measuring the photon number, it is simple to prove that for S ~ 1 and d ~ 1,

IV. Phase Operator

s!fs = 1 + log S

187

(IV.18)

It is then derived that there is an entropy gain by thermalization

of a coherent state .t1SN

= slfs - sgs

"J

log VB, as

sgs ~ !(1 +

log 27r8) for the CS. The Poisson distribution being approximated by a Gaussian distribution of mean 8 and variance 8. In order to do the same calculations for L184>, we use (IV.13c) to obtain, for v --+ 1,

s~s ~ log 271'"(1 - v 2 ) ~ 2 log 2 + log 71'" -logS

(IV.19)

so that after the phase normalization, one obtains 8~

1,

~1( 7r) ' S C4> S ~ 2 1 + log 2S

The used approximate distribution fits very closely the dominant central curve of (IV.13c') but does not reprodu~es the side oscillations. Remark that the entropy for phase measurement is a decreasing function for both variables 8 > 1, d > 1.

Finally, notice that the sum .t1TS = s~s+slfs ~ 1+ log 71'"+2 log 2 >

des, as required by the entropic uncertainty relation (1.2.3). Another method that yield similar conclusions is based on the expansion of (I.2.10a) with respect to the thermal parameter () (square root of the average photon number), using results of the thermofield analysis. However, these results are valid for a small () only. This method has recently been used to study the case of a weakly squeezed state. It is concluded that some information could .be gained for high values of the squeeze parameter [16].

188

Introduction to Photon Communication

In fact, the point of view of truncating the Fock space, does not differ from the Susskind-Glogower model for the statistical properties at least as long as the states are normalizable. However, other questions should be clarified [11,12], such as the restriction to physical states (physical states are those where all number moments are finite (n q )

< 00 Vq), the nullity of the trace

of all commutators, the fact that exp(iid)IO} = exp ((d + 1)80 ) Id} where d is supposed to tend to

00

(a state of infinite energy), and

the fact that normalization of the phase states for finite d and d

~ 00

take place in different spaces.

In order to avoid the difficulties due to the fact that the phase state is defined only on a subset of the entire Hilbert space and in order to have a superposition of all states (n

~

(0) as for the def-

inition of coherent states, a recent modification has been proposed

[20] in the form II/>} so that A( r)

rv

= A(r) L:~=o ex~~}) In}

r and assuming both r

~

which is normalized

1 and

{l ~

1.

From another point of view, a different formulation which eliminates the asymptotic procedure by using the QND method [18] has also been proposed. To measure the phase--operator, defined by an analytical representation in the unit disc [14], it has been suggested that the system of interest should be coupled with an apparatus and that the measurements should be made on a more acceptable physical observable of the apparatus, e.g. the angular momentum system in a finite dimensional Hilbert space. The Hamiltonian for the system commutes with the angular operator of the apparatus and it is therefore a QND operation. The observable to be measured is the meter observable which is diagonal

IV. Phase Operator

189

in the basis where the momentum is a pure differential operator. An initial state for the apparatus has also been proposed and the corresponding reduction transformation is found.

Finally, it has been concluded that the operator representing the phase of a quantum field cannot be separated from the measurement process. Thus, for a measurement scheme, a corresponding quantum phase operator 'can be defined (see also (15]).

To briefly conclude this section, let us consider the performances of some phase receivers in information transmission, the performance being characterized by the cut off rate criterion. As already noticed, the cut off rate criterion is not as meaningful as the channel capacity criterion. This is partly because, for noiseless M-ary channel, different receivers, namely o.d.o. and number operators, lead to equal cut off. However, it is useful when

(i) comparing the performance at a first encoding approach,

(ii) comparing the performance for the same receiver but for different types of input signals.

In the following figure, it is clearly observed that the phase receiver is the worst compared to the o.d.o. and the homodyne. It is interesting to notice that the slopes of the variations versus the average signal photon number, are equal for weak S.

190

Introduction to Photon Communication 10°

R o (nats) o.d.o. q. 10- 1



(a)

s Fig. IV.2a. Cut off rate performance given by (11.3.3b) for phase shift keying channel with the (CS) model of phase with different receivers: the curve o.d.o. is plotted by using eqs. (11.3.18), the curve q.c. is obtained from (111.3.3d), the curve quoted from (111.3.3f)-(IV.13b), versus S the average signal photon number. 0.6r------r------.----~--~-~-----.,

4J

0.5

+

0.4

0.1

: ~.

. . . . ~s.s. ····os····

~.

0.3

0.2

~

R o (~ats)

~

;

.

)

) .

.

.

.

.

;

+

; ~

[

!

T····

!(b)

~

~

.

.

s

OL--_----'-_ _---L.-_ _--L....-_ _. L . - - _ - - - L_ _----I 2.5 o 1.5 2 0.5

Fig. IV.2b. Cut off performance for the same channel in the case of three different phase models: (CS) coherent state, (TS) thermal state and (SS) squeezed state with f = fm(S) (see p. 136), versus S the average signal photon number.

IV. Phase Operator

191

From the above figure, it is noted that constructing a receiver with a squeezed phase state is interesting for S :::; 3. However, in that range of S, the resulting performance are not greatly improved compared with the coherent phase state, contrary to the case of the photoncounting processor. Calculation of performance for higher values of S and analysis of the results will be done in a future work.

IV.2 References [1]

[2] [3]

[4] [5] [6]

[7] [8] [9] [10]

[11]

[12] [13] [14] [15] [16] [17]

P. Carruthers, M.M. Nieto: Review of Modern Physics, 40 411 (1968). R. Loudon: The Quantum Theory of Light (Clarendon Press, London 1973) J.M. Levy-Leblond: Annals of Physics, 101 319 (1976). K. Kraus: Physical Review D, 37 3070 (1987). D.T. Pegg, S. M. Barnett: Phys. Rev. A, 39 1665 (1989). J.R. Shapiro, S.R. Shepard, N.C. Wong: Phys. Rev. Lett., 62 2377 (1989). M. Charbit, C. Bendjaballah, C. Helstrom: I.E.E.E., IT-35 1131 (1989). W. Schleich, R.J. Horowicz, S. Varro: Quantum Optics V, Eds. J.D. Harvey and D.F. Walls, (Springer-Verlag Proceedings In Physics, 1989) 41, pp. 133-142. J.A. Vaccaro, D.T. Pegg: Optics Comm., 86 529 (1989). J.R. Shapiro, S.R. Shepard, N.C. Wong: Coherence and Quantum Optics VI, Eds. J.R. Eberly, L. Mandel and E.Wolf(Plenum Press, New York 1990). H. Ie Pas de Secheval, C. Bendjaballah: Quantum Aspects of Optical Communications, Eds. C. Bendjaballah, O. Hirota and S. Reynaud, (Springer-Verlag Lecture Notes in Physics 1991) 378. M.J.W. Hall: Quantum Optics, 3 7 (1991). J.P. Dowling: Optics Comm., 86 119 (1991). A. Vourdas: Phys. Rev., A 45 1943 (1992). J.W. Noh, A. Fougeres, L. Mandel: Phys. Rev. A, 45 424 (1992). S. Abe: Phys. Lett. A, 166 163 (1992). V.B. Braginsky, F. Ya. Khalili: Phys. Lett. A, 175 85 (1993).

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[18] [19] [20]

V.P. Belavkin,C. Bendjaballah: Quantum Optics, 6 169 (1994). K.R.W. Jones: Physica Scripta, to appear, (1994). S.L. Braustein: Phys. Rev., A 49 69 (1994).

Conclusion In this Introduction to quantum communication, we insisted on two points. It is first stressed that in general the problem of optimal quantum measurement subject to some criterion, cannot be reduced to a problem of a classical statistical reception. As a second point, it is emphazised that the solution to a problem of quantum detection depends strongly on the minimization of a specified average cost, whereas the maximization of the mutual information transfer depends on the given input states, leaving the parameter that characterizes the redundency, free to vary. These two problems are defined in different contexts and generally have different solutions. In this essay, no attempt was made to establish a correspondence between these approaches. The two points must however be more elaborated. First, in a few situations there is some analogy between the quantum and the semi-classical treatments and useful insights can in-

deed be gained from the classical methods. Secondly, in classical theory, the maximization of the mutual information associated with hypothesis testing results in determining the threshold of the test, derived from the cost matrix. Thus, the rate distorsion which can be seen as the minimum quantity of information transfer subject to a given error for making a decision, is a special case of interest. The results for specific cases have been presented together with some elements of a quantum theory. A more detailed analysis certainly warrants further research effort.