. : http://qilab.phys.msu.ru/papers/diploma-bargatin-ru.pdf
: Mon Feb 4 18:39:11 2008
: Mon Oct 1 19:52:37 2012
: IBM-866
r 3 qp Y6 XW 76 2 % i X# $h V6 a V6 g f' V6 aQ 76 0

e % # E 0 )"

R d $c $" b E 30 a` Y6 XW 76 98 V6 Q

76 98 76 54

3 ( 2 % $# " !

! $ " 10 ) ( ' &





. ..

-





2000 .

U 8 3T GS P6 I( H H H GF

R # % Q 90 76 3E C E @ @ B0 % $A )" @

@ % A DC


--2--


- . - . , , , . -, ( , ), - . , .


--3--


1 2 2.1 2.2 2.3 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "" ........ 4 10 10 12 15 17 18 19 22 23 28

. . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3.1 3.2 3.3 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .........................

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 5 31 35


--4--

1



. -- (entangled) . 30- , , " ", . , , , , , . , [1, 2] [1]. . , [3, 4]. [5], [6] [7], [8] [9]. , [2, 10]. , , ,


--5-- . . : [11, 12] [13, 14]. , "" [15]. "" , , [16, 17] -- - . , -- - (). , , . 50- , , , , - . [18], -- [19, 20]. , - [2124], [25], [26, 27] . , [28, 29] , -- . , ,


--6-- . , - [31] - [32]. , .

"" - (). (entangled) Q 1 , : Q = A B C . . . , (1.1)

A , B , C ... -- A, B , C... Q. . , . e. , , , , . 1 - 2 , -

:
EPR

1 = (| 1 | 2 - | 1 | 2 ) , 2

(1.2)

| i , | i -- i- , , .
1

-

[7, 33]


--7-- , . e. -, . , (1.2), . : (), (), () (), .
EPR

, () (), 1/2. , -- , 1/2 -- , . e. , . , - 0 0 0 0 1/2 0 |2 ) = 0 0 1/2 00 0 0 0 , 0 0

cl = ^

1 (| 2

1

|1 |

2

|2 + |

1

|1 |

2

(1.3)

{| 1 | 2 , | 1 | 2 , | 1 | 2 , | 1 | 2 }. , (1.2), 1/2 , .
EPR



cl . , ^

EPR

( ), , . cl . ^ , -


--8-- (1.2): EPR = | ^
EPR



EPR

1 | = 2 ( |

1

|1 |
1

2

|2 + |
2

1

|1 |
1

2

|2 -
2

(1.4)

- |

|1 |

|2 - |

|1 |

|2 ),

.

0 0 0 0 1/2 - 1/2 EPR = ^ 0 - 1/2 1/2 0 0 0

. , 0 0 , (1.5) 0 0

C (1.5) (1.3) , . , " ", , (1.3). , , EPR ^ , cl -- . ^ , , . , , , , , . EPR ^ cl . ^ ^ ^ - cl EPR , (1.3), , , , [3]. (1.4) : S ( EPR ) = Tr E ^ ^
PR

log2 EPR = ^

0, EPR , , ^ , S ( 1 = Tr2 EPR ) = ^ ^


--9--
1 S (2 = Tr1 EPR ) = Tr 2 ^ log ^ ^ 1 1 22

^ = 1, Tri 1

i- , ^ 2 2. 1 - , (). (fidelity) , . , , -- , . , . , , , , . e. . - , , . , , . Mathematica. - [34], .


--10--

=
o n
- - ^^ ^
(1) (2) -

. 1: .

2



2.1
R , - R, ( , , , , -- . . 1). [23]: ^ i^ ^ = - Heff , + t h
(21) (ii)


i,j =1,2

(ij )

2

(2- ^ ^^

(i)

-- i- ,

-- , eff

^ , . H

, -

: ^ H = - ^
(i) z

eff

2

+ (i) ^

(i) +

i=1,2

lkj

e d



<
f

(j ) +

(i) (j ) +

+ 2+ ^^

s

r mj s

hgf ih

i

g

f

q p

w f x u t w gy f f fy xw vu t

;

- - + ), ^^^

(i) (j )

(2.1)
(12)

=

+ h. c. .

(2.2)


--11-- = L - A -- , ^
(i) x (i) (i)

-- i- ; x , y , ^ ^ ^

(i)

(i)

(i) z

^

(i)

=

iy , i = 1, 2, , , ^ .

1 . 2 g , f :
(12)

=

(21)

= g , = f , =

(11)

=

(22)

g f , , , - , R [23]: 3 2 -3 g = G ( ) = 3 2 +3 cos sin cos + 2- 3 cos sin +2 3

f = F ( ) =

[e1 e2 - (e1 eR )(e2 eR )] -

[(e1 eR )(e2 eR )] , (2.3) [e1 e2 - (e1 eR )(e2 eR )] +

sin cos sin -3 + 2 sin cos - 3 2

[(e1 eR )(e2 eR )] ,

ei -- i- , eR -- R, = kA R -- , k
A

= A /c -- , -

A . (2.3) , , - R (. . 1) . 2. = /2, . : | |
1 2 g e

= |e 1 |e 2 , s

= |g 1 |g

2

-- |
a

=

(|g 1 |e 2 + |e 1 |g 2 ) |

=

1 2

(|g 1 |e 2 - |e 1 |g 2 ) . -

, ^ Heff , / -


--12--

. 2: g () () f () (b), , , - (. (2.3)). |
a

- = (1 - g ()) / -

|s + = (1 + g ()) (. . 3). : | g , |e , | , |
a s



-- . -

, . - ,
(1) (1)

+

(2)

,

-

(2)

.

2.2
, R , ,
(1)

=

(2)

= ,

- (2.1), [25] .

}| { }| { }| { ~ }| { sx P g x } g } w v u t x

s zy

xy

z t s ty





d d d





x v } }



D


E




--13--

|Y
v

|Y
v
: . 4().

c
鶷

|Y


. 3: - . .

4 Ne = N a = , ( 2 + 4 2 + 22 )2 + ( 2 + 4 2 )(f 2 + g 2 + 2g - 4f ) 22 (2 2 + 8 2 + 2 ) , Ns = ( 2 + 4 2 + 22 )2 + ( 2 + 4 2 )(f 2 + g 2 + 2g - 4f ) Ng = 1 - N e - N a - N s , Ns (, ) , = 0.5, (2.4)

-- , R -- , : =
(1)

Na (, ) . 4(b).





|Y

= ei (2) . -


--14--

D


opt

E

= | ( ) | , (2.5)

. 4: (a) (b) = 0.5 ( ). ( ). (2.4) . 4, : = ( ) /2,
opt

, -- . | | = |/2| , ,

, , ( ) , .



f xx }} }

}



f } }}



d ?


--15--

opt

. 5: (a) (b) ,
opt

= 0.5. -

t = 0, | g . (. e. ) .

2.3 ""

opt



opt

, "-

" (fidelity) , . . , ( , ). . 5 , (2.5) = 0.5.
opt

, -

, N
max s,a

(. . 5) ( ,

-). . 6(),



?

?

IH GF ED 06 4' C1 BA 0' 9@ 98 21 (' 76 5) 43 21 0) (' & ? ?? ?? ?? ??

?

? ? ? ) ? ? ? ? ? ? ? ? ? ? ? ? ?

" ! # $ !% !

Ir g Iq g Ip g ih g PQ V fe Bd i 7 0 2 5 7 E e 2 4 0 I ` Y X W UQ P TQ P SQ P RQ P PQ P

PQ P ts uv R Q P wv xyx S Q P u uy T Q P yu x f edU Q Pc (b a v

PQ V

D

E


--16--
l Eh j k i Eh j j i Eh g l Eh g n Eh j m Eh g k Eh j r
R

D
v p o

E

. 6: (a) . (b) (2.6) . 3/4, -- 3/2, -- 1. , -- . , . . . , ,2 "" , . , , ( (2.5)), . ,
2

, , (2.3) , -

[23]; R, a , a -- ,

s

q Ep r

r

q p o

u p o t p o s p o

j

j


--17-- . , |
a



|e ( (2.4) , ). , , |
g

|

a

,

, . , . 6(a), |
a

,

. opt

. 1/- = 1/ ((1 - g (R)) )
s

1/ |

a

|

[

[35], () ].

2.4
, , (, , [3]).
1 2 , -




--18-- .
3

[3], : P P (0, 2 /3) + P (2 /3, -2 /3) + P (0, -2 /3) 1, (2.6)

diff

diff

diff

diff

(1 , 2 ) -- -

, .. n z , -- nz , , 1 , -- 2 OX ( , n z ). , , (1.2) |
a

(2.6) 0.75. ,

(2.6) , 180 OZ 4 , |
s

|a , , -

. . 6(b) (2.6) . , < 0 , 0 0.4 0 0.5 , , 0 0.75, .

3



, , , 3

, -


4

, ,

, ,


--19-- . , , ( 1/ ), . , , . , , . . , , . , , , , .

3.1
, , , 5 . , - , , , . ,
5

-

[24] V- [37]


--20--

|

|

. 7: - - . : i^ ^ = - Heff , + ^ t
i,j,k=1,2 (ij ) k3

2

23k ^ ^^

(i)

i j , k 3 3k -- -, k - (. . 7);
(i) kl

^ |k |l i . H : ^ H n ^
(i) k eff (i)

=
i,k=1,2



(i) ^ k 3 nk

+ k3 ^ 2

(i) k3

-- |k i ,

|k |3 (. . 7). , - g g
k3 k3

fk 3 :

(12) k3

=

(11) k3

=

(22) k3 k3

. g = F (k3 ),
k3

k3

= G ( k 3 ), f

= k3 R/c. 13

-,

=

-

C



d

d

(j ) k3

- 3k ^^ ^

+ k 3 k 3 ^^

=
k3

(21) k3

f

k3

23

= , g

7x w

{z y

g

}

~} |

W

W

g

|

(i) (j ) k3

- 3k k3 , ^^^

(i) (j )

(3.1)

--
eff



(1) (2) 3k

+ h. c. ,

(3.2)

k3

-- -

= gk 3

k3



k3

= fk3 k3 ,

(2.3):

13

=g

23

= g,

13

=

23

= .


--21--

|

. 8: - -. , , - , , - (. 7). - R, . , . g e |k k = |k : |s |a
kl 1

, a s --
kl

=

1 2

(|k 1 |l 2 - |l 1 |k 2 ) k , l = 1, 2, 3, k < l.

. 8. ,
(i) 13

23 . -

(i)

. , -

e

e

|

|V

|D

|





|V |D
2 7

|V |D

|k 2 , k = 1, 2, 3, (|k 1 |l 2 + |l 1 |k 2 )

=

1 2


--22-- |11 . [39], . e. , - , , . |11 , , |2 |3 -.

3.2
, , . e. , - - , "" [38]. , -, . , - , , |a
12

|s

12

, -

, , - , -. , - - , , 2. : -, 2 .


--23-- , , , . , , , (, , ). :
k3 k3

= 0,



k3

= /2,

| k 3 | =

(i)

|| (1 + g ),
(1) k3 (2)

(3.3)

-- (

= exp(ik3 )k3 ),

, |1 |3 , |s |2 |3 , -- |s
13

( - -

, |11 ), , 12

,

. -:
(i) 13 13

= 0,



23

= k 3 ,
(i) 23



k3

= -/2, || (1 - g ),
12

(3.4)

| | =

|| (1 + g ),

| | =

|a

.

(fidelities) , 2.3, . - 9 9, . . 81 81. , , .

3.3
() [40]


--24-- . . "" - [41], . , . , [42]. . -, - "" ,



13



23

-- , 13 (t) 23 (t) -- -

0 13 (t) 213 H = 0 223 23 (t) 2 13 (t) 23 (t) 0,



(3.5)

(. 7). , , |1 . "" [41] d = cos |1 - sin |2 , (t) tg = 13 (t) . 23 (t)
13

(3.6)

(3.7) = 23 , -

, , ,

-,


--25--

W W p Q

c m



Q
{ m



. 9: . . , (3.5) , |3 , -. "" (. e. ) (. . 9), , 0 /2 , |1 |2 . ,
-

k3 dt

1, k = 1, 2, -

[40], . e. (3.6) . [40], , :
13

= 0 exp

(t - p /2)2 , 2 2 p

0 -- , p -- . - -




p







23

= 0 exp

(t + p /2)2 , 2 2 p

(3.8)


--26-- , , , [43]. , , , - , |s
12

|a

12

. -

, c , (3.2). - , , . , -

, , , . , . , (. e. ) , , , , |a |s
12 12



. , , : = 0, = , = /2, | k 3 | | |, (3.9)

13

23

k3

(3.8) (. . 9), , |11 |s
13

|s

13

|a

12

, ,

,


--27--

12

. 10: (a) |a

-

f . (b) 13 . , p = 0.1/ . |a
12

.

23

= , -

, , , [17]. , . . |a
12

. 10(a) -

0 p f = / ( , ). , f , , 0 p 5, - (3.9) ( ), (3.6). f , 0 p 5, . . 10(b)



? ? C C 0 { { 7 2 { f 7 { { 2 f 2 e e G e e { {









0 e E 4 i I 0 { 0 0 4 C 0 5



0

e e

e e

e e ~ e

D

E


--28-- 13 ( . 6(a)). , , , , , p 1 , . . , .

3.4
, [39], . . , , . , , : = 0, = /2, | k 3 | | |. (3.10)

k3

k3

, | | , (

) , . , , . 11(a) ( -- . 2.3 -- ). , |a
13



, , (. 1) . , , , , . , , ,


--29--
! ? ~? ? ? ! '% " $! % ! &% $f e &d c b
(1) 13

. 11: (. 3.4). , , -- . - ( = exp(i)(2) ), -- .

, , , . , .
k3

= ,



k3

= -/2,

| k 3 |

| |,

(3.11)

. 11(b). |a .

, , , , , . , , , , , . , , |33 ( 2.3 , -

0) #( (





$! # "

? ?



! % "

3) #2 2 1

tf ts s b

f $e d g b T SR Q

978

&6 5 4

f d qh g b

W $V U

GF E DC BA @ f d rh c b P 3I H

f e ih c b

f e ph g b a `Y X

D

E

wf vu u b


--30--
w B t 3 w 0 i & w q q q 0 q x
12

. 12: |a (b).

(3.1 : (a) -

), , - . , 2.3, , , . -, . . , , , 81 81. . 12 , (3.10) (3.11). , () , , [41], -. - (3.1) , |1 |2 -

t q w pp ' tw i ix } s~ i~ pu |t #p px w iv ix s} |u &{ zp yv tp qx vw &t Sp v sr pu 3t sr qp o x y By x y Dy x `y By x x y x

By x By x d By x e By x fgd h By x i l k j x Dy m n

s B t w 0 r w w 0

' & p ' t # & w & r | t q # i & | & q | # p ' q r & B D

' ' ' '

B B B B B D

D

E




--31-- -, , L
jitter

= ^

12 i,j =1,2

( 2zi) ^ (i) z

(j ) z

- zi) ^^ ( ^
(i) 1

(j ) z

- zi) zj ) , ^( ^( ^

(3.12)

12 -- ,

=n

-n

(i) 22

-- -

- i- . , . 12, |a
12

, . e. - ,

, , . (3.10) (3.11). , = 0.001 . , 0.8 1.

4



, -, . , , (., , [44, 45]). , , ( ), (. . 13), . , .


--32--

.....

. 13: . 1 [46], , . [17], . [29, 30] , , (. 14(a)). ( ) . , , [30], "" . (. 14(b)), , . 14() , Z. ( , ) . , . , , -


--33--

. 14: (a) , . (b) . [48]. , , , . , , , "" [30], , (2.3). , , , [47]. , - , [49]. , , . , (-


--34-- ), : E =
j =i

^ | j | H I |i | Ei - E j

2

,
t

(4.1)

^ HI -- . (4.1) [47] U - 3 c 2 Il (r), 2l3 l (4.2)

dip

l -- l A , -- , I l (r) -- . ( ) Il (r) = I0 (sin2 kl z + sin2 kl x + sin2 kl y ), (4.3)

kl -- . , U - 3 c 2 I0 k 2 (r - r0 )2 . 2l3 |l | l (4.4)

dip

:
2 loc

=

2 kA kl

A | l | , 3 I 0 m

(4.5)

m -- . [47]:
A

= 2 /

A

= 800 ,

l / = 103 , m = 100 ..., loc : I 2 l 6 /2 . 2 4oc Al (4.6)

loc


--35-- , , l = A ,
loc

= 0.1 (

) 600 1 2 . (4.6), . [30] , , .

5



. - - . , , , , . -, -. , ( , ), ( ) .


--36-- , , 0.8 . , . , , -, , .


--37--


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--38--
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