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ISBN ???

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3 3 ! ! . 1, S (X Y ), 3 ! 02 , "- X Y ( . . 9 , "-, 0 " 3 1 3 3 ! X Y 3 1 . . ), ! 0 S (X Y ) = S (X ) + S (Y ) ; I (X : Y ). , "- " . 1, " " 3 . 1 . 1 . !. ? ." 8 " "

:

D- H (p) . 9 3) ) * 1 . ) 0 6 H (p) 6 1. / 3 !02 " . ," *! 32 , . . *! 3 !3 1, ," * 3 0 2, ! . / 1 , Y = y 3 , X = x .3 p(yjx). 9 3 S (Y jX ) 3 ! X X S (Y jX ) = ; p(x) p(yjx) log p(yjx) = (2.3) x y XX =; p(x y) log p(yjx) (2.4)

2

2.2.

2

22

(2.8) pnp (1 ; p)n;np = 2;nH (p): > 1 " * ! 1, * ! ! ) 1 " ! * ), 3 " 3 ) . 3 1 * *2 " . 4 . . \ *

, " 3 ! 1 ) ! 0 2;n(H (p)+") 6 p(3 ! 1 1) 6 2;n(H (p);"): (2.9) V! 1 " 3 . 1, n . ), * ] ), *.0 3 0 3 ! 1 1 10, * 18 ), " 1 ; " 3 !

* 18 . n ." , 1

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3 . 1, 3 3 3 ! 1 ), 2nH (p), . 5 ! ,

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3 1 nH (p) * . , ] " ! 1 *2

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p(jm ; npj < n") > 1 ; (2.10 ! m | !-, ! 2 3 ! 1 n .

). ( * 18 n !- m *! ! np , ! "0 3 0 n. 3" , ! " 3 ! ) !-

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! n np ! 1 p ! -! = np(1 ; p) C (n m) n!=m!(n ; m)!. /8 3 " ! 3 ! 3 " .1 , . " " (1): ! 3 ! n . ) X * !" 3 1 3 . nS (X ) 6 n * 4 ) " ! . , " ! 3 ! *

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2

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24

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*. ", . (6), 3 ! 02 3 0 , "-0, 3

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2

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2.4. , !"# $ % .

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*. ": u ! u0 = u + e, ! e * . 8*, . 3 3 * 8 ! 3 " * . 3" , 3 * * u = 1001101 ! u0 = 1101110 " * 1 u0 = u + 0100011. ! C , 302) 8* 3 ! " ,

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1

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3 ! 8 s, * !" "8 410 ! . 8 * * " 3 8 0 ! ) .! . ) " ! 1 3 ! " * " ) . W 2 ", ! -

3.6. ( !

8 s 3 "1 . *+ " , "- (3" , s / L3 + L), ,

02 9 "

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*! 1 ! " . 5 1 * 3 2 3 1 8 , . . . 1, ", 1 . NP # 0 * " .! 8 " * 1 3 . 3 "1 ". 5 ! , P 2 NP , ! , 2 0 .! , !2 NP !2 P ( . . NP 6= P ). ! 1"

3 ! ! ! * ! . , 3 1 0 1 ." 1 2 " , 2 )! . 5! , " 1, 3! 9 . " ! , . . ,.

3-3 , 3 "310 , 3 1 "8 410 3 * " 1 0* ) ! ) "310 93

-1", 3 "1" ." ! ". /" 3" " 3 !!02 ) 8 0 .! . ! . " : .!81 " ( . . 02" 3 ") " x * !" 3 ! 1 ! . "

). W x ", * " "-* * 1 8 " , " 1 ) . / " 1 -0 ! 3 " x 0 * 18" ". / 9

3 ) " ! 8 .! . . * . " ! 8 3 # (Menezes et. al. 1997). 5 . . 8 s 3 ! s exp(2L1=3(log L)2=3 ), ! L = ln x. ( 3 1. 2 02 !8) ! 1 1 ) 3 ! " ) , 2 . 130 ! . (Crandall, 1977), . . L ' 300 3 * 8 s 101 9 .! 8 ", " 8 * " . (3 * , ) 1012 3 -) ! * 42 !). 5! , 1 *+ " , "- L ! . 8 s,

, 3 ! 1 ! s 1025 , .! . 8" ): 3 " " . 1 ) 8 .)" " , * ! 8 * !" 1 * ! ) "310 " .. / 9 "

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3 ". ( ! " ! ! ) 410 3 ., 2 " ! ,

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1 " 2" " " " " ". 5 ! , 3 " 210 " * * . 8 1 3 " " . ( 1 2 ." 1 ! " *2 !, ) 3 ! " "8 410 . 8 1 0 * 3 0* ! !. 4 ) " !

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1, ! dT ] | 3 "8 T . V! 1 ! "8 TH ! dT ], ! , "-0 "8 410 T , ! !. 5 0! ! :

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3 1 .8, ) : E = M s mod c, ! s c | 2 1 3 ! * * 18 - , " * 1 *2 ! 3. 8,

*2 3 1 3 ! . : E t mod c, 3!02 . " M . ( " t ! 3 " 210 s " ) c (Schroeder, 1984). 3 c = pq .! 3 . ! ! !

3 p q, . 1 " 3 1. 0, ) .! . c. ( 9 " ! 3 , 3 -*!1 " . 1 " c, 9 3 1. 1 *! ! ", *! 3 * 3

1 !

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40

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4

!: * " 1, ." ] 0 - *, * ( ! " * 3 !3

1") 1, ji " ) " ) ", 3 ! *+ " , "-, ! 2 ) ) 3 " ). / " A B . *2 ) , "- ), 3 9 " *+ " , "-, ! 2) B , ." , 1 . * 1 - A. 2 " " 8 3

3 ! ) , "- , 3 9 ",

! . ". 2 1) . 93 " EPR 3 ! 8 !" .1 ". /3 * )! " (1964, 1966). ( !3 ", ] * ." 0 02 3 1 . ) O A O B 3 xOz . .1 " ! ." +# -#. 93 " " ", 1 3 ! ! ." ) ! .1 sin2 ((A ; B )=2), ! A B | " ! 10 Oz " O A O B

. 5! , 2 3 * 3 ! ) , . . ." ! ! ) - A B , 3 ! ) ) -, 3 ) .1 *! 3 ! 3 3 ! !, A = B , 3 ! , A = B + 180. " , 1 3! .1 , 3" , sin2 (60 ) = 43 , A ; B = 120. D )" (1982) 3 3 ! 2 1 ., 3 ., 8 -, * 3 " 1" " 3 " ", 3 A ; B = 120 23 . . 1 Bell-EPR 3 . 3 ! 1 ,. ." 0, . 8"0 3 " 210 "310 .! : ! ! ! ! ! 2 ! !

3 ! 3 !0 . A B , !/ ( " . 1 " 18 " 3 ! " ! !"

") 3 10 0 , A = B + 180@ 3 3 0 , A = B , 0 10 * 18 ) 70%, A ; B = 120. ?3 " 1 3 ! . 1 * 3 ! 1970- 1980- !. / ! 9 3 * 3 ! !

42

? " ) !- ) ) , "- (Schumacher, 1995). 5! * " " 1 " !" " (3" , 3 ; 21 ), * ! ) " (". . 12), ! , 3 3 ! * *+ " ) , "-, 8 * * ! ) : ,

" ! n * , ! 2n -" 1* 3 , " *. ", ! ! 3 2n * " ) (3 "", n * " 3 ! 1 ! 2n . ).

3 ! * *! ! * 3 ! * .! 5.6. ! " .3 1 ! 1 ! * fj0i j1ig. * *2 2n 1 ) " * 1 .3 fjiig, ! i | n-* ! . 3" , " * " 3 ": fj000i j001i j010i j011i j100i, j101i j110i j111ig.

5.9. .%

) ) , "-, 8"

*. " 3 . 1 ! ) , "- " 3 " ." .. .1 " 3 -1 " .! J. Mog. Opt., " 41 (1994)@ *. 1 3 3 , ! " (Bennett et. al. 1992), F " (Hughes et. al. 1995), D " (Phoenix) 4 ! " (Townsend, 1995), ! " (Brassard) 3 (Crepeau, 1996)@ ? " (Ekert, 1997). :3 (Spiller, 1996) ! *. 3 . 3 ".

5

4

H

I j0ih0j + j1ih1j = 9 1 (5.2 X j0ih1j + j1ih0j = W (NOT) (5.3 Z P ( ) (5.4 Y XZ (5.5 1 (5.6 H p (j0i + j1i)h0j + (j0i ; j1i)h1j : 2 / , 1 ! " 3 ", ! ) 0 ! * " * 1 . 3 ! " -* "1 ; ! 1. / ) , "-, 3 !" 02 ) 2 1 ! ) , ! ) 02 ! * : ) 0 02 T-W# 3 - W (NOT)@ ) , "- 2 * ! -* ) . ) ) " 3 *. j0i j1i * . ) . X ) ", 3 1 3 " ( x. : ! " 1, " ) fI X Y Z g 33 ) 3 " 0. . " ." 3 3 ! 3 !" , " ! j0ih0j I + j1ih1j U 1 2 & , & 3 . 4 H .

! = !t. " * 1 .3 ! P () j0ih0j + exp(i)j1ih1j. 3 . ! 9 " ) :

( 3 - ! * " .0

" " ) "# (Deutsch, 1985, 1989). 3" , * 3 ! . ! ! j0i ! j0i j1i ! exp(i!t)j1i , 3 3 8 " t * .! ) ) ": P () = 10 e0i (5.1

5.10. *

5.10. *

5

! I | ! -* 3 - 9 , U | )-*

! -* ) ) . 4 ) ) . controlled U #, 3 1 .! ) ) I U ) * " (j0i j1i) 3 * . 3" , ! ) ) controlled-NOT (CNOT) .3 ! j00i ! j00i j01i ! j01i (5.7) j10i ! j11i j11i ! j10i: V! 1 ) * 3 ! # . 3 -0 W, 1 " , ! 3 ) * ! j1i. ) 3 1 ." ) *- ! ) . 53 - ( !" !") controlled-CNOT ! " jaijbi " * 1 .3 a ! a b ! a b, ! * . 3 -0 0 02 T# (XOR). ( 9 ) 3 !) ) . XOR- ) ". 3 - *0 * 18 * . 3" , 3 - (AND) . 3 ! " -* ) controlled- controlled-CNOT, " ! 1" * " 3 . ! 3 - W, 1 ! ! ! j1i. ) ) . 1 4 ,, (Tooli, 1980), ) 3 ., ) 1" ! * " ). ) ! ) jaijbij0i a ! a b ! b 0 ! a b. " " ) 3 3 -0 (AND) ! !" 3 " * ", ) * ! j0i. V! 1 3 * * !" ! , * 3 - - " * ! , " *. ",

-" ) 9 0-. :" ! 3 ! 1 "*- ) ! .- 9 " ," 3 -) ! " , 1 ) .! ). 18 " * 3 ! - (Barenco et. al., 1995), * * " *2 3 " !

/ ! " (Vedral et. al) " " (Beckman et. al., 1996). 5*2 .! ) 3 ! 1 ) "

46

. 3 3 ji * !" , 3 ! 1 3 " 9 0-, 3 " ) : U (jij0i jiji, ! U | ) 3 9 0-. W 9 3 ! 0* , 3 U ! .

, ! 1 , U (j ij0i) p= j ij i ! j i 6= ji. 5!

! j i = (ji + ji)= 2 3 ": U (j ij0i) = (jiji jiji)=p2 6= j ij i , ! 1 , 3 - 3 . ? ! 3" " 0* " 3 ! " " !

(Wooters, Zurek, 1982). : ! " 1, !) 3 # U 3" " " " ( ji j i 8 3 !

3" ). , 3 1 !) 3 !, ! . " ! * 1 1": hj i = 0 3 , 3 . , 3! " ! " . 8 ., 1. 10 . 1, )-* *) 3 U * 3

/ 9 ! 3 3 ) , "- ) ! ) , 3 ! * , 3", ! 0 3 0*

3 ! " ) , "-. 53 - CNOT XOR .

. + * ! . ? " ", 1. 3 1

3 ! 3 , 3 3 ! ( . . 3 3 , "-, .02 ! .)

5.11. - . ( * ! !

4

* 1 .3 3 ! " 3 : 3" , X1 H2 XOR13 ji ! ji | . * , ! 3 3

3 .0 " * , ) .! ) 0 . 5!

3 ! 1 1 ) , ! *

. " .!" " ! 3

, 3 9 " ." 0 !"" ), . " ) ) 10 (". . 8). 3 !02 " *! 3 1. 1 " 9 !"".

5.11. ,-( * ""

5

( . " !02 ! , * !!! * " 3 1. 1 ! 3 ) , "-. 3" , ! 3 ! * 00101 ] 3 * 3 1 * , !2 j00101i. *, 3

*2 , " . 1 . , "-0 3 ! " ." ! * 3 *. fj0i j1ig ( . . .1 3 " * ." " ) ). .1 ." ! . 3 ! 0 !0 * . ( 3 ! ! * 3 ! , "- * " * ! " " * . 4 3 1 3 !3 ", ] * " 0 .- 3 0 3 * , !2 j00i + j11i ( 9 " " p ! 3 .3 *! " 3 1 9,,- " 2).

9 ] * ! * 1 1 .1 ! ! ": 3 !3 ", 2 " ) , .- 3 3 * 3 !) . * ! ] * (". . 9). / 9 " ] "

*2 1 * , "-0 ! *-

5.12. 0

(28) 0 3 -" 3 ! p ) j0i j1i,p

! ) j+i (j0i + j1i)= 2 j;i (j0i ; j1i)= 2. 4 " " 3! EPR ! ! 1 " !", ) ) " 0 1. ) 1 , * *

." , - EPR * 3 * .1, 1 1 3 8 * 0. ? , 0

!1 ! 3 0 (! ) 3 3 ) 3 1 3"0 . 3 -1 ) 1 . , * 3 1 9 , ." ", *, .! * 18 , , ." 3

. " *.", " ! . 3 ! 1, ! 3 EPR *. " fj0i j1ig *. " fj+i j;ig. ] " * * 3 1 3 " 0 .1 3 ! " 3 ! 3 EPR ) *. ." *

, 3 ) 2 ." .

48

4

- * , !! !

5.13. ! !

3 ! " 3 ! 1 ! * ( . . 3 !

) .- 3 ) 3). : / (Wiesner) !) " ! . 3 " ! "#, 3 1 ! 3 ! ! * ] 81 ! ) * , . . " , 3 - 3 ! , "- .! ) ! * , 3 81 !" . . ) " ! 3 !02) , : ! ." 1

) j00i + j11i j00i ; j11i j01i + j10i j01i ; j10i 3 ! ! ! " " * 1 * 3 3 ! " 3 -)

!" * ". ) * ) . *. " 3 1 3 0 * 10 -0 Bell-EPR ." (Braunstein et. al 1992). j00i+j11i ] " 3 1 0* . ) *. 3 !

.! ) " 02 * !" . 3 fI X Y Z g ( 1 2 1 ." 3 , * .! ) *! 3 ! 1 ! * ) , "- ( 3 ! * * ! 3 ! 1 " . *. ! !) * . ? " ! 1 3 !

.! ) 3 * ) " XOR ." .1 0 2 * (target bit). 4" *. ", * j00ij11

) j01ij10i. 3 ! . 3 3 .- ! 3 1. 1 ! 8 * 3 *. ]!" H , . " 3 . ." .1 . 4" *. " * ! . 3 , "-0 ! * . ( ! 2 " " 3 . , " ! " ! .. 5! , !) " ! * 3 .2 .: 3 1 , "-0, ! 20 ! * " 1 " , -*

*! * ", 3" * 3 " " ] ). / *2 3 !

! 0, 3 " 2

" ! .! . 5 3 . ." .1 " ! ) , "- ), * " " , "- " .- 3 (Barenco, Ekert, 1995). T* ! " -

) 3 Mattle et. al. (1996).

5.13. " ! "

aj000i + bj100i + aj011i + bj111i: (5.8) ( 9 ] ." 3 *. 3 ! * , . .

." . ) * ! * . .- 3 ) 3. 1 ! 9 ) 3 - 3 . . 9b. ( 3" ] ) ) XOR ) ]!", ! ." * , 3 " !02 : j00i(aj0i + bj1i) + j01i(aj1i + bj0i) + (5.9) +j10i(aj0i ; bj1i) + j11i(aj1i ; bj0i): / .1 ." , 3 !" ] ), 2 ." ! ! . ." 3 ! ! ! * . ! * 3 !0 *,

3 " 210 3 ! , ) . 3 fI X Y Z g " * !" 3" 1 " * ! , * 3 " 1 aj0i + bj1i = ji. 4" *. ", * ! * ( . .

(5.10 ! Tr | ) ! 3 , | 3 , 3 ! 02) 3 1 ) ) ". ? * !" 1 " (1), 02" 0 9 30 ; . ( !3 ", ) 3 " X 3 ! . 3 ! ) p(x) W " ! jxi, 3 ! P . " X , " - 3 x p(x)jxihxj, 3 " jxi *. 1 ! *

1". U 3 . 1, 9 3 S (p) ) *2 ) , "- I (X : Y ) !

X .1 ." " Y . 3 ! * " " " , * !" ! 3 ! ) " q, .! " ) " - ) 3 . ! 3 3 ! * " (

S (p) = ; Tr log

( , * ! 3 * , * !" 3 . 1

! * ) " ) ) , "- ! . 1 * 3 . (Jozsa) ;"

(Schumacher, 1994), 31 * (Kholevo 1973) T (Levitin, 1987). ( ! " ! . 1 * !" * 1 , 3 .020, ) *+ " , "- " 3 1 3 3 ! ) " Q. ( " " ) 9 3 , 10" (Vo Neumann)

5.14.

0 , "-0, 10 0 "), ! * 3 ! 1 ]. : ! " 1, , "- 3 ! * 1 " , . ] ( ) " ). " , , "- 3 ), 3

1 ji | 9 3 3 * ]. ] " 3 -# *2 " ! 8 . , . ! 3 - 2 ) 3 " .! . 3 " ".

( !3 ", ]

3 ! 1 * ! * ji. W ) . * , 3" ji = j0i, " 3 ! 1 * 3 " 210 ) , "-: ) *, * ! j0i. : ", ]#. 5! , ji . , "

3 ! 1: 0* ." " . 1 ." @ " , ] " 1 , ! , * 3 ." 3 ! . 5 ! ,

! ) 3 * 3 ! 1 * * ji 9 3 1 " ,. ) * ( . . 9 ", -* 2 ), *

3 ! !0 0 " 3 1 . / 0* " *! 2 3 ! ) ". / ! . 9 ) .0 3 1. " ! ) 3 - (Bennett et. al. 1993, Bennett, 1995). 3 ! , *! " 3 1. 1 .- 3

, "-. ( !3 ", ] * " .- 3 3 * j00i + j11i. ] ! 3 ! 1 * ! * , !2) . " ji. / " " *2 " " 3 1: ji = aj0i + bj1i, ! a b | . 9,,- . : ! 1 , 1 * :

5

5.14.

5

50

5

n 1) 3 ( ! #) " 0-* " * . 3 !. " 3 ! 3 . 3 3 , " ! ! # n " q0 , 3 ! * " q (". . 9c). ! " q0 .! " - ) 3 0, " 3 - 3 ! 3 8 . 8 ", " - 0 !

3 " - . U ) 3 ! * " - 3 3 ! : p f ( 0 ) = (Tr 1=2 01=2): (5.11) " " 1 1 3 ! " - ) 0 3 3! " - ) . / , ! * " - 0 3 ! 0 : jihj j0ih0 j,

1 3 ! " " 8 . " 3 ": f = jhj0 ij2 . / ! " .! .0 3 ! " " 18 3 !

10 3 ! f = 1 ; " 3 " 1. . 1 ! 3 3 ! 1 )#, 3 ! ) .! 2.2. 5 1 ! 3 " " " !" ", )! ", 3 n " 3 " 2n -" " 1* " 3 . 5! , 9 3 , 10" S () < 1, 2 1 ( . . 1 " !- 3 !

* 18 " n) , 3 0* ) .! ) .- , 3 )! 1* 3 . ;" . 3 ., ." 1 ! 3

3 !3 2nS(). : ! 1 , !

3 ! ) , "- * 1 nS () * , " * ( . . ," ." 1* 3 ) ! * ) " ) ) , "-. , 3 - ! ! ! * 3 #: . , "-, . ) 1 3 ! " . .1 , 3 ;" " ., 3 ., 3 1 *2 " 0: * ! ! 32 )

1 3 ! " " ). / , " * 1 1. / , ! 3 ! " ." 1, .! ! " 0 3 ) , "-.

52

5

! *. 3 ) , "- 3 ", " 3 " 3 ,. ) .! * 3* ) * / (Wiesner), 3 ) 3" 1970 ! (Wiesner, 1983). / 9 ) * * . ! 3 3 1. 0 ) " ! 8 .! 3 , .2 ( . . 1 , "- " .. . ! .! " ! 1 !02 3 !.! : (quntum ke distribution) *+ ! ! ) *2 2 .! (bit commitment). / *2 3

3 3 ! 0 *! ! . D-0 * "

-, ! ] ! 3 1 -* 8 3" , 3 1, , * * * , 3 . 3 ! ". ( 9 " * " . 1 8 ] 1 3 .! , ", " 3 ! ), 1 " .!) " ! ! ,- * .0 ", ] ! .3 1 *"

8 , 3 " 1 ),, ) 3 3 ! *. W 3 .! .

, * * . 8 ,

*2 " ! ),. 43 ) ) 3 2 1 .*

5.15. ! * !

! # ! ! #, * !" ! ! 3 ! * ! "3 ) , "-, *0 1 . * 3 . !8) ! 1 . " 2 1 ! 3 - ! 3 1. " ,

. 5! , ""1 ." "

. . ,.. 4

3 * "

3 ! (Cleve) /- . (DiVincenzo, 1996). " 3 " , "-, !

! , , ! F,,", 8

! ) . " ;" 1 (Neilson, 1996) , 0 . ) , "- )# " ) *2 ) , "- ! ". ?

. 1 *2 ) , "- " ! .- 3 " "", " 3 " 3

" 3 1 - Bell-EPR.

5.15. " !* "

3 3 1. * 3 .1 . 5! ) * 10 , " . 1 .1 ." * 3 " *0! . * " " 3 . * 3 !"" 3 * ", 3 1. "" W ) ( ! . 3 ! "" * ") ! 3 ! 0 , 3 , ." * 3 !02 3 ! *. / 3" 3 " . W

3 ! *., 3 1. ") ] ) 3 9 " 8 * . 5! *., 3 1. " W ) 3!0 *." 3" *. 4" *. ", W . 1 3 n * , 3 ! ] *

! ". " , 8 8 ) 3

, 3, 3" , * j+i " j0i

3 ] ). ( * 8 " " *! .1 ." * 3 - * 3 1 , .! ] ). / .1 W 8 n4 * RQT. 4 3 1 ] * " . 1 3 W 3 ! " ) * n2 * RQT 3* 8 . ). W *+ . 3!0 , ]

" * 1 , 3 !8, 3 1 1 , W 3 !8, * n2 * 8 " ", (3=4)n=2 ' 10;125 3 n = 1000. ? 3

* *.0 ) 0 . 3 3 * ", 3 1 W "

3 1. 1 . 3 !8 (3" , 3 1 * )@ " , ! 3 3 8, 3 " .* . . * . /"

. 0 , ! " . 8 * 3 !0 , ] * 3 1. 1 " ! 3 , 3

8* 3 25%. ( !02 ** 0 ! 8 . ( ) 8 .0 * ! 8 * 3 ! " 3* ) 3 3! . ) * ) * 3 ! 1 )@ ! " " .

* - 10 ! 3 1 3 W ) ! 3 1 ) , "-. " 8 . ! 0 ! ! ), " 1 8) 3 ! , ) . 3!02 . ) 3 1

. ! 1 8 3 " ! *. !02 ) ! : ] 3 ! * * , 3 .!

*2 " ", " *. !) * * 3 ! . ( 1 ) 3 , * 3 ! ! @ ! 3 1 " ) *. , 3 " ! .! . 3 , *! ! ) * ) ), .0 02 ) ",

." 93 " 1 ! . 1 .22 1 3 - !: ! . , ! 1 83 " 8 . " 1 ". : ! ) ,! -1 1 3 1. ) ! 3 1 " " ! . 1 .2 ., " 9 ) * * 3 . ! . 1) *+ " * . : 3 . ! . 1 .22 1

2

3 3 ! 0 . 5! , ! * ! . .22 1 . 3 ! " " ! 302 ) ! 0 ,- * (Mayers, 1997, Lo Chau, 1997)@ ! 1 " ! .1 ) 1. ? ! . 1 ! . , . " 3 1. 1 - .- 3 . ) 3 3 ! 0 " ! ", " ! 3 ! ) 0 ! 3 , 3 1. . 3 ! , ] *, !2" . 1 " , * !" * 3 1 .1 ! ! ". ] 3 ! * 2n * , !) . ! "-*!1 ) * " : j0i j1i j+i j;i2 * ." 3 * 3 ) * " *.: fj0i j1ig fj+i j;ig. ( 9 ] * 3* ( . . * . * 9 ")

*20 ! ! *., ) !) . 3 1. ! 3 ! ." * . 5 3 ! 0 , ! " ) * 3 1. ! *.. 0 3 3 0 .1 ,

02 1 !" ". : ) , 3 8* , -, ] * " 0 ! , 2 . n * (3 ! 1 ! 1 1 j0i j+i . 0, j1i j;i . 1). * . (raw quantum transmission RQT). 2 % . 5 & & , # &

* .

5

5.15. " !* "

5

54

5

0 . ( 9 " . W 0 " 180 . 4" *. ", " 3 1 ) 0 , ) 3" . n4 * , 3 9 " * 18 ) 10 . W ! " 8, 0 " 10;6 * (Bennett et. al., 1992). /8 3) 3 ! ." ". ) 3 ! ! (Ekert, 1991) . 3 1. " 3 EPR, ] * ." 0 3 ! ) . ). : - 10 0 3 !8 " * !" , .1 , 3 1 -) Bell-EPR. :2 3 "2 3 3 ! 0 .0 ", 2 " ! " . . / " " 3 " 93 " (Bennett Barssard, 1989) * 3 . 1 " !, 3 * . 1

8 . ! .! F (Hughes et. al, 1995), D (Phoenix) 4 ! (Townsend, 1995) 3 ! 0 ! ! ) * , ! V*! (Zbinden et. al., 1997)

*2 * 3 8 ) 3 ! 0 23 " 3 ! " ""- " , 3 !2 " 3 ! . " G . * 1 3 . . "31 , . . , ! 2 ! " 0,1 , "31. 4 ) .) ) 1 * * !" ! ,

* ""1 . 1 1 3 "31, ! 2 * ! , , 3 1 3 ! * "31 " 3 . ! 1 * , " *. ", 3 . 0 3 !802 "

1 ." ". / ! ) " * ! 1

8* * 1,35%, !

." 3 . "

* 3 .2 3 . : 1 3 ! ! ! 1

.: ." U\- 3 * ! ) . ), ." " \\-. , " 9 , "

1 ! . ( ! * 9,, 93 " 1 " ! 3 ! " !02 .! .

56

( . " " 8 ." 3 " " 3 ) " 0 ) , "- | " "310 (QC). 5. " 1 *. , 3 2 " "310 .! . " * 3 ! W " . (Ekert, Jozsa, 1996) - (Barenco, 1996) :3 (Spiller, 1996) " *. " " 3

3" "310 . 5*. " " ) /- . (1995) ; (Shor, 1996). / 3 0

!1 ) "310 ) " 2 81 , " " 93 " , 1 . ! ! , "1 " . ** ) , " -. / 3 " 210 ! 3 ) " . > { 410, " ) .! 4. 531 * ) (1985, 1989), !0 * 3 !+ " " "310 : ) "310 3 ! " , 2 . * , ! 3 3 ! !02 3 -: 1. !) * " * 1 3 ! "-* .

j0i. 2. !) * " * 1 ." 3 *. fj0i j1ig. 3. 1) ) ) ( " ) ) " .! ) 1 0* 3 !" * . 4. : * ." " 3 ! " 8 . 3 *. ). 3 . 3 ! , *! *! 1 !1 )8 ", ! ! "310 . 4 " ! 1 ) ). / ) 2 3 ! 1 .! ) " ) " " * (.! 1: * ). T )

9 "310 3 " ) 3

6

1) ) ) " 9 " 1 ) , . . ) , ) .! ) . "*- * , " " 1 ! ) 0*

! ) . 5! , . * 3 ! " ." ) ? 9 3 * 1 3-3" ) " ( ; ! ): 3 1 9 0- ! , *! !

.! 1 "310 * ! n * . 3 ) .! 9

" 3 . 1 ! ) .! ), 3 1 " 3 , , ! 1 , * " . 5! , 3 . ) 1985 . ! 1 3 ) " * 1 1". * 1 ) " 3 . 1 !02 " 3 " 3" . " " 3 ) : V ( ) controlled-NOT# ( XOR), ! V ( ) | ) 3 . 1 2 ! * , . . =2) ; ie;i sin(=2) : V ( ) = ;iecos( (6.1) ;i sin(=2) cos(=2) " 3 . 1, 0* ! " - ." n n, " * 1 *. 3 " "* !-* ) XOR ) 2 ) ! * . 4" *. ", ! 3

3 -) " " 1 1. U .. 1, ) V ( ) ! " 1 * " ) , 3 1 3" 0 3 ". 5! , 3 ! " * ! 3 ! - 1 . ) " 3" )

6.16. ' ( *

5. "81 " "310 ", * !" ! . 1 1, . . 3 . 1, ,-

. " > { 410. . 1 . ! 8 " 3 *

1 3 . / -3 , 0* )

) " 1 1* " 3 . 4" *. "

" * 1 3 ! !

, 3 " 210

* . / - , 9 0- 0* ) ) " ! 3 *. , " *. ", " * 1 " " "310 , ) 3 * .! 1

! 3 *. 3 . 1 )

10. (-31) 3 * . U ) " (1997), !

3 ! , 1 .! , ! 3 !

8 ! . 8 , 3 ! 0 3 ! ! / 3 3 1 " "310 , 1. 3 !

"310 . 5! , !1 )8 " *! " 1 * .! , " 3 . 1 8 ! . 8 , * .! , * ) "310

*2 3 " 210 3 -1 3 !. ! 9 * , .! ) (Deutsch, 1985). " 8 . ,

1 8 .! ! " . 1 (Nielsen) > (Chuang, 1997 " 3" " ) 3 ., 3 " 210 " " 3 1 * " 02" ! " , "- ) 3 "" ), ."

6.17. 8 9: { )(" *

!" . ", " 3 1 3 2 ! * . 5! , * !" 3 1. 1 3 ! * ! " " ! 1 ) " | 3 " "* 3 -) 2 XOR, " 3 1 3 -0 " 2 , 020 !

" 1" ) ". 53 3

! * 1 ) " ) (Dentsch et. al. 1995), T )! (Lloyd, 1995), /- . (DiVincenzo, 1995a) (Barenco, 1995). ! " 1, ! * ) !

! 3 ). " 3 9 " ) ) " 2" " 3 ".

! 1 ! !, " "310 ) " 0 ." ! ) * , 3 !2 3 ! ". ( 9 " * , 3 . !"" ) (. 8, 12), ."0 ." 3 . / ." 3 * ! " ! ) ) | 3" , " !

" (Margolus, 1990).

5

6.17. 6 78 { )(#*

6

58

6

3 ! " 9 " 2 1 ! 3 *. , "-. 5! *! " ) , ) "310 3 " ) ", ! ) 02" ) . 4" *. " 3 " 210 ) 3 "" " 3 ! 1# 0* ) " (0*0 3 ! 1 1) ) 3 1 ) "310 . . " ) "310 3 . 1" " " ". ! " " ! 3 * 18 ) . 3 , *02) . " 1 ! 1 10. 53 "310 , . 8 , 0, ! 3 3 ." ) 3" ) " * 1 ! 0*

1. , * ( 9 0- . ) 0 ,. ". 53 "310 *! ", ! " * ) ! * 1 3 1 ! 3 " ) 3

. ) 3 3 !" " !8 ! ), *! " .! 9. ( * !" * 3 ! * . 1 ." 2 1 3 "310 .

60

( " *

!" 3" " "310 " - -* ! ". " - 2n -" " 1* " 3 " "310 * !" 3 1 ", ! 2" 3 ! 2n "3 . ) - " "310 * 81 n * , ! * 9,, "

. * !" *+ " 3" . 5! , *2 ), ) "310 9,, !

7.18. ; ! 4 <

5*2 . , "310 3 * 3 ! ", ! ! 3 * 3 . , ) "310 " * 1 .! ", . 8"" ! "310 . ) 1 , 3 1 ,.

! , " " 3 1,

1 " " , , " * ,. , " 3 1 *!2 ", "

! 3 1 2 1 .! . 8" 3-3 3 " 210 ! 1 * 18 ) ) "8 410 5! , * 3 . .! 3.2, 3 ! ! 1 * 1 8# !

* # 0 0 0 1 , " V! , #

. ), 3 " . 1 3 . 8"". \ * , ! 3 , 3 8 " 8 " .! (3 0 " ") " 1 ." 3 . " ", .! " * 1 ! " "310 3 3

" ! ) * , " * 1 8 3 " 210 "310 .

7

2 " " *. " 1 81 " - 3 - ( !, ! 1 " 3 " . F 1 *, * ) "310 3 1. ! 8 * *2 .! @ * ! . , 3 .! , ) "310 * * 9,, , . ! . 5! , , , 3 ! * .! 3-3 2 0 , . 1" " ,. . 5 * 18 ) 3 3 * 3 " ! ) * . !8) ! 1 " 3 3 ! ,-

!" . " . ( !3 ", ,- f (x) | 3 ! 3 ! " r, . . f (x) = f (x + r). ( !3 " ! ,

,- f (x) " * 1 )! 3 ! " x, " , . 1 . , N=2 < r < N ! -* N . ( !3 , 2 3 * 3 ! 3 ! ,- f (x), ! " ! " 3 ! 3 " 210 "310 . ) f (x) ! 3 ! N=2 . ) x, ! . x, 3 . ,- 0 3 1 (! ,-) * !" ! " p 81 O( N ) . )). ) " ! 9,, ", 3 -

(7.1

x=0

!X ;1

jxif jxi:

(7.2 3 ! " . 11. V! 1 * !"

" 1 !020 10 * 1: . ! 8 * ) ! . ,- f (x) ! ! = 2n . ) x. ? * . * , ! 93 -1

p1!

! ! = 2n. ( 3 ",

!", !

! ) " 3 *. D1 (". . 10). / jxi * . , 3" , j0011010i, ! 0011010 ! .31 - x. / . 9 " *. fj0i j1ig " " 1 1) *.#. ( 3 "310 ! * ( , ." , *. 1 ) 3 1. 1 ! *. ". V ", ! 3 3 *. Uf jxij0i = jxif jxi " x y 3" 1 ) . : ! " 1, ! 3 *. " * 1 ! ", 3 1 ! jxij0i 3 10

! " 0 jxijf (x)i , ! 1 , 3 - 0 * "". 4 3 1 3" 3 *. Uf 0, 3 " " " (34), 3 "

x=0

p1! X jxij0i

!;1

1 3 -) 93 -1 . log (, "-, * !" ) ! 3 ! N ). /8 . .! " * 1 8 " "310 3 ! " " ! (". . 10), 3 ! ;

(1994), 302 " ! :" (1994). 8 .! " "310 * !" 2n * , O(n) * | ! .! *

3 , ! n = 2 log N ] ( n] * . *)8 *I 18 - x ). / * ! ! #, !) . ! 3 n * *! " *2 1 ", x# y#. !) . . 1 .! j0i ( . . n *

! j0i). , ! " . * 3" " 3 -0 H , 3 !

" - 9 0- ". ) "310 ! 3 1 " -", ! 2" 22n 9 " , 3 9 " 3 -) (" , ) 93 -1 . n. / 0

!1 9 " ) 93 -1 . ! 3 -) "310 ,

2 " 2n-" " 1* " 3 . 4" *. " ) "310 " 1 9,, 0 " -0 0* ) ,. ) ". 5! " 3 . 1, ) "310 9,, 3 " - * 18 ", ! " . , 3" ! " " 1" ." ! ) ", 2 9,, " 8 (Lloyd 1996, Zalka 1996, Wiesner 1996, Meyer 1996, Lidar Biam 1996, Abrams Lloyd 1997, Boghosian Taylor 1997).

7.19. =* . . =* > 4 "

7.19. :* ! ! -. :* ; ! .#

7

62

y

5 " ", (34) 3" " ! ) 3 - !, 3" ) 1 " 0 j0i. 5 ) ! ) , * !" ) ! .- 3 *. UFT " * 3 *. D1 (Fast Fourie Transform, "., 3" , Knuth, 1981). / ! ", 3" ") " " * ." .* 33 " " (Coppersmith, 1994) ) " (Deutsch, 1994). W

. " 1 4 &. & . , .

jfe(k)j =

n

1 k !=r (7.6 0 1 . 3 0 . 11b. V0 1

x 3 1 ." " ." 1, 3 . !=r. 5 81 3 ! 1 3

! r. " " x = !=r, ! | . . W " 0 *2 " , 3 x=! "

! * , " *. ", 3 ! 1 r. W r " 0 *2 " 1, " 3 * 18 . r, * !" 2 . 3 1 8 ". U 3 . 1, 1 3 8 8 .! 3 3 ) * 18 ( 3 , .! " 18 ) log r, ! *. 1 (Ekert, Jozsa 1996). ) " ! 3 ! ,-, 3) 8 , " 1 9,, ", 3 *. Uf , 3

" 3 ! 0 . ,- f (x), ". 5*2 * !" ) 3 "1 , 9 3 -1 . n. 3 ! 1 .! 3.2 9

2 5 , # & 6 & , 7 * # , 8. # 7"" ', , 7" " 9 . , &

& 6 & # (Barenco et. al. 1996).

!

(7.5

) W . (1996), - (1996)2. 3 !

(36) 3 *. 0 UFT , * !" ! 1 3 2 02 .! 3 !3 ", ! ! r * . . . M = !=r. , ! , 3 . * !" 1 ! 3 1 ! (Shor 1994, 1995, Ekert Jozsa 1996). ( 1 * 18 * !" *2 1 y *! " 1 1 x, 3 ! " " (36).

." n, 3 ! " 2 )8) 3 .". (/

*. 2 2100 , . . " . * 18 ] ! , 3 - ). F 2n . ) ,- f (x) "- " 3 0 # " , 3 ! " " " (35), 3" ) ! 3 ", 0, ." . ? . ",

!02" 8 " ", ! ." (3 1 " *.) y " 3 1 81 ! . f (x)1 ( !3 ", 3 9 " 3 " . ,- f (x) = u. : y 3 * jui, *2 *! 3 ! 1 M ;1 (7.3) p1M X jdu + jrijui i=0 ! du +jr | . x, ! f (x) = u j = 0 1 2 : : : M ; 1. " ", 3 ! 1 ,- f (x) . , x 3 3 .- ) M ' !=r ) 3 . x, . 3 ! " r. : ! " 1, " 2 du " . ) x . u, 3 ) 3 ." y. 4 3 1 * !" 3 ! 1 3 ! 1 x. 9 * !" 3 ! 1 2 1 3 *. D1 ." . 3 1. " ! 3 *. D1 3 !02 ) ! ) 3 - ! ) !X ;1 ei2 kx=! jki: (7.4) UFT jxi = p1 ! k=0 !=r X;1 Xe UFT p 1 j du + jri = p1 f (k)jki r k !=r j=0

7.19. :* ! ! -. :* ; ! .#

7

64

r

" 0! ), ."02 "

", 3 " 3 3 " "I . 3 1 ! " 3 ! 8 . " 3 3 ! ,-. 3 !. ! 8 8 ) 3 ) .! : 3

.3 3 !

) *. ! (). \ (Grover, 1997 " ! !02 ) .! : ! , ! 2 "

.3 ) fxig. * !" ) .31 xi = t. 3" " " 1 3 " , 3

(3 " * . ). ! ! . 1,

" ! 3 " 3 N .3 *0 ! " N=2 8 . p ] " \ , 0

!1, * 81 N 8 .!

. ), 3 3 " ): 1. " , ! ,

, 1 8 ! ), 3 ) .!, 8 "

7.20. =* ?

, : 3 ! " "310 " 3 ! 3 " , " . 83 | ! - , 9 " ! . ]. . 11 (34){(38), " . 1, * * ! (35) V! 1 1 ! 1 * 3 " " * , 1 * , , " ,

, -. ( ! " .- 3 x y, 3 ! " " (35), ! . ,- f (x) . " " x. D # .0 ", 3 ." y .- 3 3 . .! 1 x j ( (36), ". Jozsa, 1997). V0 1 3 *. D1 " " 1 , - . ), !2 x ( ! ) !,- ) 8 ). ?,, , - " 3 1. 1 ! ) 3 " 210 , ! " , . . ) ) * 10. : ! ) , " 1. 1 93 -1 * 18" " . " ! ) 02 ) .- 3 ".

, 3 !

* 18 n, " ! . 8"" . 8"" .! ". 3 3 * !" " 1, " 02 * 18 . .! . " , * " .! 3.2, " * 1 ! .! ! 3 ! 3 ) ,-. ? , 8 . ! , * 3

*+ ! ; " (1994). 4" *. ", 3 ., .! . " " * 1 . 8 3 " 210 ! 1

"310 . / ! " 3 ! " ,- " ! f (x) = ax mod N , ! N | , * . 1 " . V a * 3 . 1 , 3 " a < N . 531 9 " 0 0 , " 3 . 1, 3 0* " a 3 ! r " ", ar=2 1 " *2) " 1 N . 4 3 1 !) *2) " 1 ( ), ." , 1 " 1 N ) " * 1 )! 3 ! " " ?! ( 300 ! . 9., "., 3" , F! (Hardy) ) (Wright, 1965)). 9,, ,- f (x) 3 1. 3 02 . ! ! (3 " !0 N ): ((a2 )2 )2 : : : V " ! 3 ,

02 ! " . 0 a, 3 " 0 . ( 1 ) , * !" ! .- " ; , 3 U9 " (Miquel et. al., 1996) / ! " (Vedral et. al, 1996) " " (Beckman et. al, 1996) ! 3 ! 300(log N )3 ) . 4" *. ", ! . " 3 ! 10130 3 ! 1 ! " " ! 8 , * !" 3" 2 1010 ) 7 * "310 3 3 0 # 1 U\-3 . (" ! , . .! " "310 , " . 1, !) " *! 3 "2 " 3 ! " ". 5! , 1 ! . ! 260, .! " 1 . 8" ) (". .! 3.2). : ! ) , " "310 ! 8 3 * 81 "1 . * 18 " . :2 " 2 " ! 1 ." 3 . " ) . 3 ) .! " 3 ! 3 ! ,- 3 3 5 ' log 60 . , # &.. '.

6

7.20. :* !

7

66

7

! sin = 2 N ; 1=N . ( 9 " 9,,- 3 3 ! " " 9 " 3" 1 * 18 . , " 9,,- 3 4 <.. # NP!, ',

-& '.

p

! j | " .3 t = x, 0 3 ! ) . 1p j(0 )i, ! sin 0 = 1= N . 8 ) 3 3 .- ). V " * !" 3 ! 1 3" 1 3 S , ) ." . 3 ! " 9 " .3 3 3 ), 3 1. 1 3 *. " D1 , ." 1 . 02, " 02 ) " j0i 1 * 1 * " 3 *. 0 D1 . / .1 !

3 -) " *0! 1 . 1) 9,, , -, ) 3 ! !02 " 3 *. 0 UG ji = j( + )i (7.8)

8 1 ! ) " ; 3 . 0 " ( exp(2(ln N )1=3 ). 5 * 180 1 3 * 18 " (N ' 1016) , 3" , .! ) ! 8,- .8,

*2 ) (Bassard, 1997). / " 1 !02 : ! ., " \ 3 "1", . .p ! ) " " * 1 * , " O( N ). !1 3 ! * , * " \ " 3 1 !02" *. ": ! .31 " " i, * !"

! . 3 ! 1 ! .31 ), 0

) . " ", ! 2 1 ! ) 3 S ), S jii = jii, i 6= j S jj i = ;jj i, ! j | " 3 -1

3 ! " 9 " . 3" , 3 ) * !" 3 ! 1, i 8 " ) ) 1 ) .! 4 " " 3 ! 3 ! ,- ( (34)) !) " ! 3 ! !

! 3 3 .- 1 ). 53 ! " X jii (7.7) j()i sin jj i + pcos N ; 1 i6=j

p .! , " * 1 . 1# N=2 3 -

68

6

1 02. : 1 ! " ! p .0 3 3 *. UG m ., ! m ' ( =4) N . / ! " !

2 . 3* =2 , 3 9 " 3

* 0 " jj i. ( 3 ) .1 " ." ( 1 8* O(1=N )). : ! " 1, . 1 3 ! 1 . m (9 * ! ) " (Boyer, et at., 1996)), 3 1 3 3 ) 8 .! . D (Kristen Fuchs) !) " ! 3 " , : 3 " 2 0 3 1# 31) " ! 3 1#. / ", 8 " 8 " 3 .! 3 1. / 3 " 3 , ! ) 8*

". 3 ! "

1 3 3. : 3 " 210 " 3 0 1 .

*. " ! ). 5! 3 2 ! " !

". (Kitaev, 1996) 3 . 8 .! . " 3 02 ) 3 ! " " !, ,!" 1 02 ) " ! ; . U ! * 0* .

. (1997), ) 3 . *2 1 " , 3 *. D 1 . 4" *. ", " ) 3 "" 3 3 . 5! , 10 " . 1, .! , ! ) "310 3 !3

1 , *! *" , ! 1 , * 18". : !

, 0* .! , 8 ) ) , 3

1, " , 3 ) ) " , * 1 8 3 " " 1 3 ! 8 . / 9 " " \ "

3 ! 1.

7.20. :* !

V 3 ! 50 . ,. 93 " * 3 . * 9 " 3 -. 3" , 3 - W (X ) 1 , ! ) 3 ! " ! !" 9 " " j0i j1i. " 02 3 ! . 3 - XOR " * 1 3 ! 3 ") 3 ! ) " . 5! , ! " * "310 * !" ) " !

3 "0, * ) " " .! ) 1 )-*!1 ) ) ) , !

0, * 1 3 ! * ) , "-. ( "

1 . U ! 1 .! 1 ) ! " ". ? 1 * " ! " . " 3 . ! , 3 3 0 " " 2 "310 . 5! , 30 , - 9,, ) ! 10, ) 3 ! 1 3 .-, 0 3 " . ! " " 1 * 0 . ), ." ! ) 02 ) ! ) ! ! , ! , 8 0 . / ! 9 " 1 02 ) ! 3 ! , ." ! ) )

1 1 . ." ! ) 3 ! 3 ) ! . " 3 ! 3 !. 5 1 3 1, !

" 1 ! . j0i j1i, " 3 * 0 *1" (3" , . 3 ! ). * !" , * 3 ) 3 3 .- j0i + j1i 1 ,. ! ). 5! " 9 " " " ! 3 " 8 " ".

,

8

7

U ! ) 8 3 0 . 12. W 3 ! * 3 " ) : (Steane, 1997). 3

3 " 2 )0 8 (# * 0 ) " (10;8 ( 3 ! " "*- 3 " 9 3

). T . 2 3 3 ! " ! ) - 3 " ! " 3, ! .

* 81 ! . !) . " ! 1

! " . 3 ! 1 ), 3 ! , 3" ! " " ) ( " . 3 ! * ), 3 3 1 3 " 3! "

1 ). 5* . " ! jgi jei 5 1 " 3 ! 0 ! * . ! 3 . ) . ) 3 ! " " ! " "

02 . 3 - 3 .

3" 1 0* ) ! -* ), !-* ) , . . ! .- 3 ! * !" .! 1 ." ! ) " ! ", * 3 " 9 . 5! , ! , 3 1. 1 !

8.21. ; ! .$

!8) ! 1 2 ! 3 3 ", 3

* . 1 ! 10{40 * . ( " * 3 ! " (Cirac) V " (Zoller, 1995) .0 3 1. ! " , 2 ! ) 9 " ) .!, 3 " 2 0 8 ! " . / " * 3 ! \ 8 , 1! " (Gershenfeld > " (Chuang, 1997) ! " (Cory et. al., 1996). W 1 3" " ! *+ " ! " . . 5* 9 3 ! 30 ! ) 2 * 18 33 ! ), .* 93 " 1 " !. 5!, * 3 ! , 02 93 " 1 ) (Lloyd 1993, Berman et. al 1994, Barenco et. al. 1995, DiVincenzo 1995) . ! , " " !, 1 93 " 1 . "" ! (Privman et. al. 1997, Loss, DiVincenzo, 1997) "

2 "" *)8 " *!2 ".

8.21. < " -%

.0 1 " * 3 0 0* 3 *.

*2 ! 3 * ), 3 !+ " " 310 (.! 6) * !" .! 1 1 ." 1 . 8 3 ) .! ." 3 ! " "

! 3 ) . ! . 8 ) | 3 " 210 ." 1 " !: 3 !, * electro shelving. / ! " ! 0 " 2" " ", . * " . 3 ! !!- 1 33" , . 3 * "" " ) ,.. " 9 , ! " 3" " ! * ! 93 " 1 81 ! ! / ! " 93 " * 3 ! " ." , .! 3" ) ! ! ! 2 8 (Monroe et. al., 1995). \ 93 " 1 1 " ! ) 8 . 0 ! - 3

! 8 (! "3 ! ) ) 1), "

" ! ! ! . " ! ) " ! ) - 3

), *!02 ) .! " 3 " 8" 9 ! (Steane 1977, Wineland et. al. 1997). . , 1 " . 1 . U 1 ! , . 1 ." " 3" 1 ! 100 ) 1" " * . 3 U ! 1, . * ! 1) 3 ! * 3 . ( 3 1. " ) .! ) )

3 ! . 5 ! , 3 - , ) " ! ) 8, ! " ! 1 *+ " 3" , * !" ) ! . " .

. 5! , * * 3 . 1 3 1 * " * 1 * ( 4 ! 10) " * . ", ! 1 ! ) 3-3 ** ) , "- / .! 9 *! 3 . " !, 3 . 0 . 1 1 ) 3 -).

." ! ) @ ! *+ V 3 * * 18 3 - 1 1. T . 3 1 9 0, "31, 3 9 " 3 ." ! ) ", 3 ! *" "31". D , ." 9 3 ! ",

3 ! , - 3

! 1 (9,, U * (Mossbauer)). ! ) - 3

" ! , 3 " 2 3 -, .! ") 8 ) (. !

0 . " 3 " .*! (, "#) "1 " ! *) - 3

. / , *! " ! jn = 0i " .*! jn = 1i ) " !. : - 10 3 , 3" ,

3 - controlled Z# ! ! x y " * !" 1 ! jn = 0i. "31 . ), ! ) 02 x, 3 ! !02 " 3 !: jn = 0ijgix ! jn = 0ijgix jn = 0ijeix ! jn = 1ijgix. 4" *. " - 3 ! ! " , ! 3! 1" " : ! ! ) 3 - ) *" #. , "31 . ), .! ) 02 y, 3 ! 3 !": jn = 0ijgiy ! jn = 0ijgiy jn = 0ijeiy ! jn = 0ijeiy jn = 1ijgiy ! jn = 1ijgiy jn = 1ijeiy ! ;jn = 1ijeiy : / .0 .! ) " x 2 !" "31 ". 5*2) 9,, ! ) "31 !02): jn = 0ijgixjgiy ! jn = 0ijgixjgiy jn = 0ijgixjeiy ! jn = 0ijgixjeiy jn = 0ijeixjgiy ! jn = 0ijeixjgiy jn = 0ijeixjeiy ! ;jn = 0ijeixjeiy

3 - controlled-Z# ! x y. !) . ) "31 ! 1 3 3 ! 1 . 53 - controlled-Z# ! -* ) *-

7

8.21. < " -%

8

72

8

( !3 ) " ! ! " . 3 0 . 13. / ! " ) 3 - 3 ! * ) " , # ) . 3 ! ! " . 9 " "" 3 !0 ! , 3" , " ! !, , * .! ) 1 " .. 9 " 3 ! 0 ! " : ! . *! " " " " ", " . 3 !, ) 3 "

3 ) * ". U 3 " 2 *+ " " 3 , 3 " 3 ! 2 3 ! " " 3 3 ! 1 "31 *02 " 3 . 2 " " . 3 3 ). : 1 .0 ", 3 ! " ! ) " " * 1 .! , ." . . 8 8 ) - 3 1. ) *+ " ! , ! 2) 3 ! 1020 " ! 4 3 1 ." " ." 1 ! 3 " 3, 3 1 !) *02) " ) " " ! .! ! " " 3 . V! 1 . 1 1 ). ! " ! " , ! ! " , 1 1 " 3 , 3 9 " 9 0- !

3 - # 1 * 18 . . / ! " .! 1 3 1. ! " " ! U | " ! 3

9, 3 . ) . 1 * ) 9 0- 3 , " 18 9,, .! ) ) . 5! , 9 " , . 1 3" * 18 3 ! 1 ) ) . 4 3 1 * !" 8 1 .! .! 1 . ( ! " ", *+ " ! ! "3 " , ! 1 , 3 . 3 ) 3 ! 0 3 ! 0 1-". / 3 1. " ", , 9 , 3 *. ! ! , 3 9 ", . 1 " 0 3" 3 . U - 3 ! O(1020 ) 3 ! *. 3! ! ) " - ) I . " . 1 = ; I 3 . 1 0 , "-0. " , ! . 1 " - ) 3 )-*

8.22. @ * 4

74

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8 3 " * 3 0 81 ** ) ) , "-, 0 .1. 5! , 1 ." ! ) " ! " " " !

) " ! ) 9 " 3 " * 1 3 " 3

" . . ." ! ) 3 1. ) ! " ! 3 , " *. ", 3 * 3 ! ) , "- " ! ! " " 8" 3 ! " . 3

(Cirac et. al. 1997). ( ! * 93 " 3 ! . *

!" 1 ." ! ) " ! 3 " . " " * 3 ! " (Brune et. al. 1994) 4 (Turchette et al. 1995). " , 9 " 3 " * 1 3 1. ! * 3 ." ! ) " ! " 8 ,

8.23. A 4

) ", ." 3 *. 1 3 " 2 1 3

! * "31 " 3 " *. ", " 3 . ( 9 " " . 1, 3 ! * ) 9,, ) ) "310 . ( ! * 3 , "

* 1, *. 1 . . 9,, , 3 " 3 ! " 3 ! 1 "31

2 02 ) , "-, ! \ 8 , 1 ! " > " (1997). V 1 93 " 3 1. " " ! U * 3 3 ! ) 1 3 ." 3 , ) * !" ! **

) , "-, 3 ! " ) 1" * ". 4" * . ", 3 ) 3 - , 3 02) 1" * "

) , "- *! 3 ! 1 ", * 02 3 " ! U. 5! , 9,, 1 " ! 3 * : 3" , ." 1 ." " n * 2;n . " , 2 ." , 3 1 ." ! 1 8 ! * 18 3 - . / 8 . ." 1 3 1. " ! 3 8* (.! 9) " " .

8.23. ! .

8

" *. ", * 3 . ! 1 ) 1 ) , " " ! (Pellizzari et. al. 1995).

76

/ .! 7 * " 1 ! ) " " . 5! ! 1 " .! .! , *02 * "! ) , *

! ) 3 " 81 * ! ) " " ! "310 (.! 1 0 " ! 8 .! " - ,. ".) ( . .! 93 " 1 ", " 1, !) " " ." " 1 "310 #, 3 02 81 1 " ! " * , , , 1" " ) 1. 3 " " . 1 "310 "#, 3 1 *!0 !

" " : 8 " ! " " . 1 "" 3 - ", 3 0 2" ) , "- ). " *. " . 9 1 .

" ! "" ! ) 1"? ", ) 3 * ) " " 310 , 3 ! ) .! 6, ,. 2 "". / ! ) 1 2 ! 1

) , . ) ". U ! 1, ! !

3*. 1 1 ) ! 1 ", !) " " 9 2 " ) " ).

) XOR ." ! ) ! . 1 .! * . * ." ! ) 0 ! ! ", .* *! ." ! ) 1 "-* 2 (Plenio Knight, 1996). * !" 3 " 1 ", . ) ) " ) 3 " * "

! . " 3" ) ) XOR. . 9 ! , ! 1 * " 3" 107 . " 18 " . " 130-. ! W2 3 ! . 1 3 . 0 . ,.

1 ) 3 ! !

, ! ) " 10 . 1,

! " #$

9

" (Shor, 1996) * *2 " ! . 5 " 1. . ! ) * 1 18 . ;

(1995) : " (1996). ! . " .- 3 #

3 ! " " (Bennett, et. al. 1996) ." ) (Deutsch et. al. 1996). 4 QEC . (Knill) T," (La~amme 1997), W (Ekert) U (Macchiavello, 1996), (Bennett, et. al. 1996). / 3 ! ) * 3 3 "1) 3 * ) !, ." ) T," " " (1996). \ " (Gottesman, 1996) ! * (1997) 3

*2) ) 33 , . ! 3 *. 3 * 3 ! 0 " ! (Calderbank et. al. 1996, Steane 1996). ! 3 !1 * ) 1 . " ; " T,"" " ! U1" (MacWilliams), 3 " ) ! . ( 1 " ! QEC 0 * 3" ) ." ), . 1 * , ! ! * 1 ! 1" - 10 * 3 ,- " " !. / ! ; (1996) (1994)

3 ., ! 1 3 8* 1 " 8*" ! ). " ", 3 ! * - ) 8* # ) .0 3 " * 18 , " .!0 U ! ; * * *2 /- . ; " (1996) 9, , 1 * : " (1997). T,"" (1996 3 ! ! 0 # ! , 020 " " ! " ! . W 1 .0 * 3 ! ! 1 3 ! 8 3 " 9 " ) 3 - 3 ! . 5! , 9 3 " 2 " ! * 3 . 9,, 3 1. ) 3" ( " *. " ! .- * !" " 2 ) "310 ). 3 . 3 " * 3

1" " (Knill et. al. 1996, aharonov, Ben-Or 1996 Gottesman et. al. 1996). 3 .! " ! - 8* 3 ", \ " " ". 5 " ! QEC 3*. 1 3 3 " . " " !, ) * 3 * !0 3

" 3 ! ) 3 3 " ". 1 !

1 1 ) 106 ., "

1 . . 1 ! 1 " 2 " " "310 . ! . 1 * 3 ! \ 8 (Haroche) )" ! " (Raimond, 1996). * , * T! (Landauer, 1995, 1996) ! " " 1 3 !3 ! . 3 ! * 3 ! "310 ! " (Unruh, 1995), (" (Palma et. al. 1996) > " (Chuang et. al. 1995). : 1 .) ) *. ! ) 3 * " 3 ! U " (Miguel et. al. 1996) -

(Barenco et. al. 1997). "310 ! ! ) ) .* , 3 " 1 3 " ". , * 9 3 1 * !" 3 ! * . 1 * , 3" , * " 3 0 . ) 1 3 ! "*- ) 3 - : * 18 3 0 3 0 # 0 # 3 ! 3 0 * 18 ) .202 ) 3. W " 0 . 5! ,

! ) 81 .202 ) !

: 3 ) , 3 0 1 * 1 ! 3 ! ". " . , 3 0

* 3 3" " , ! , . " 3. / . . . . . 5! ,!" 1 . ) " 0 0 3 ! " ! ) "310 . 4 " 3 . 1 3 ! , !3- " " ! ) 9 0- ). 4 ) ,!" 1) 3 ! ! 3 3 0 8 3 " ", " 0 ." 1 .2 "310 ) 3 " . ( ! 3 - "310 3 ! " 2 1 * ." ) " 3 * !

" (Berthiaume et. al. 1994, Miguel et. al. 1997). 5! 3 " 210

3 1. , "- ) " ) ! . 9

3. ! .0 3" " "" " ! 3 8* ) , "-. U ! 3 8* (QEC) * 3 ! : " (1996) ." ! * " (Calderbank) ; -

7

<! ? %&

9

78

( 9 ." " 3 " 1 3 " 3 . 1) .1 : 3 ! jesi(Ms je ia)j 3 " 3 ! " . s. / 9 " * ! " 1 *2) ! 3 " , 81 ! 3 ! ) 3

s

/ ! " " 2 * : . . ! 3 1 n ; k * , !2 j0ia. . " ! ! 3 1 " 3 " 1"#. 0* ) .! ) 3 - ! 2 3 - ! A, ! ) 02 *20 ", 20 . " qc a ?,, ! ) ! 3 3 ! A(Ms )je ij0ia (Ms jei)jsia ) ! 0* Ms 2 S . " S 3 ! * ) "

3 " 8* . 3 - ! .

* . jsia, s ! " " " 3 ! 3 1 . ") 3 8* Ms . ! , jsia 0 ." 1". 3 . 3 !3 ",

*2 3 " " (". (44)) ! 81 3 Ms 2 S . / 9 " 3 3 - 3 ! ! " " qc a " 3 ! 1 ! : X jesi(Msjei)jsia: (9.4

s

* !" 3 1: X Y Z . ? : 8* 3 * * # (X ), ,. 8* (Z ), ! " 3 ! 8 * (Y = XZ ). ( !3 ", "310 q 3 k * " ) , "-. ( 1 *2 k * 1 ji. ( " ! ) " *! "310 3 ! ! 3 !02 n ; k * j0i. 5* . " 0 " . qc. 52 " 3 -0 ! #: E (jij0i) = jE i ( 1 3 1 3 " ! ) 0 n * " qc. :

*2 1 . , 3 " " * 3 ! * 8 *

3 -)# M , ! ! 3 - . " 3 . ! " n 3 ( ! 3 !) * ), * . " fI X Y Z g. 3" , M = I1 X2 I3 Y4 Z5 X6 I7 3 n = 7 5*2 .8" : X jesiMsjei: (9.3

* .- 3 # (Bennett et. al. 1996, Deutsch et. al. 1996). 5 ! " ! .0 ", ] , " "

.- 3 3 * , . " 3 ! * . ! ) 3 3 3 " " *. ] * 30 * , . " 2 0 3 ." " 3 : 3" , * 2 3 -0 XOR ! 3 !02

. " * , . " ." .1 02) * . ( , ] 8 ! 3 - ! " * ", 0 .1 . W .1 3!0 , " . 1, * 3 ." * )

3!0 * "": j00i+j11i. W .1 3!0 | * *0 . ( ! " 3 ! * 3 . " .- 3 3 * 3 " " ,1 1 3. *! !" .- 3 " " ] * " . 1 3 ! " 3 -. 3 ! * 3 " ) (Bennett et. al 1996). 531 3 ! * ! , . 1 . , ? (van Enk et. al. 1997) 3 ! " 3 * ! ) 3 ! ) , "- " ! "", !2" ! ! ! HQ 3 . 3 ! " ! 1 3 . *2 *! 3 . 3-3 QEC. V38 " !8) . ." ." ), " " * 1 3 ! * : *2) ) * 0 ( . . 3 ) ." ! ) * 02 ) ! ): jeii(aj0i + bj1i) ! a(c00je00ij0i + c01je01ij1i) + (9.1) +b(c10 je10 ij1i + c11 je11 ij0i) ! je:::i * . 02 ) !, c::: | 9,,- , .2 3 " . ( ! " 1, ! *2 ." ! ) " .3 1 !02 " ! : (9.2) jeiiji ! (jeI iI + jeX iX + jeY iY + jeZ iZ )ji ! ji = aj0i + bj1i | 1 * , jeI i = c11 je00 i + c10 je10 i jeX i = c01 je01 i + c11 je11 i . !. V" ", 02 ) ! *. 1 0 ". ". . (43) , 2 3 8* ,

8

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80

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* ! *!0 8" 02" 3 * ". / 3 )8 " QECC, .! " 3 8 3 " "

* 18 * . V! 1 .! .0 ! ! , 302 8* (QECCs), 3 * 8*, ! ) 02 , """, t * . (

! *) QECC *! . 1 !, 302) t 8* # (t-erro correcting code#). ( )80 " ,- ! (3 ! ! * ", ; " : ") " 3 1 !02" *. " 5 " ", -3 , ) ) ! 3 " 8 * , ! ]!", 3 ! ) *- 1, " * 1 3 1. ! 3 8* 3 X . . 1 3 (17), 3 . 3 3 - . ! " 3 ! 1 3 " 1 , *! . 81 3 8* Ms, "310 ji. 9 ! , k * 3 ! " 2k ." 1 n | * ) jii, ! !

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8* 3 Z. V" ", Z = HXH . : ! 1 , 3

8* 3 Z 9 3 H ! * 3 0 8* 3 X * " 2 .

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96

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