Ð Ø ÓÖ Ñ ÐÐ ÓÒ Ý × Ò × Ù Ð ×

 ! " #$
% & ' ( !" !
# $ % $ & ' (
) # * #+ ,
+ $
"
-
+ & # ' & !
! !#$# %& !#$# ! & !#$# !( !# %& !# *+ !# , !#$#
-
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/
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012 # . 3 #. #. . !# 5
7 "
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$8 2 -
9 ,9 , 1 -
:
; < %1 = -
-
1 . 11 ;< %1 : > 9 , 12 1
1+&2 -
1+& ? @ ?@
AB
C D
,9 EE 2 -
1 . $ 12 E 1 F , -
51 ?@
-
! 1 1 9
! " # " $ %
&' ! ( &)*
-
& 1 9
. , E
' 2 %& ?/@ -
&
. 5 ! :
$
-
C
-
9 h̄/2 -
9 2 . -
C
-
1 5 .
+ , 1 $ 12 +
$ 9 ,
2 9 : -
+
,
C
2 9 & E2 -
2 -
!
2 9 2 & 9 $
-
9 -
9
;<2 %1 2 -
& 9
; < . 9
* E
G & 1 ! G2 &
σx 2 σy σz
9 9
2 ; 1 2 8 #1< F 12 &9 -
-
-
2 2 /
-
9 2 & 2 -
-
9 2 -
9 2
9 !2 2 9
82 1 , 1 9 9
1 $ E 2 -
C σz 9
2 σz 9
1 2 -
9
$ 12 2 1 1 9
-
σz 9
& 2 . -
2 . 9
2 1 , 9
1& 1
-
&9 &
.2 2 1
& E ; 1< 9 -
-
+ ! 1 2 -
2 12 1 2 -
-
9
! E 2 -
-
2 8 2 * 1 ! 9
9
9 H ! & ;
5< 1 9 9
9 H
+ -
. 2 &= 9 , 9 9
1 &
1= 2 & ; 9< 1
9 2
-
9
!
-
9 9
9 H 1 1 +2 1 σz ⊗ σz 2 1 0
0 −1
σz =
; <
,
, &:
| + + = |+ ⊗ |+ =
| + − = |+ ⊗ |− =
| − + = |− ⊗ |+ =
| − − = |− ⊗ |− =
1
0
1
0
0
1
0
1
⊗
⊗
⊗
⊗
1
0
0
1
1
0
0
1
⎛
⎜
⎜
⎝
=⎜
⎛
⎜
⎜
⎝
=⎜
⎛
⎜
⎜
⎝
=⎜
⎛
⎜
⎜
⎝
=⎜
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
⎞
⎟
⎟
⎟,
⎠
;<
⎞
⎟
⎟
⎟,
⎠
;<
⎞
⎟
⎟
⎟,
⎠
;/<
⎞
⎟
⎟
⎟,
⎠
;<
|+ . |− 9 9
2 -
2
&2 -
9
9 h̄/2 −h̄/2 z σz |± = ±|± | + − -
9
σz 9
& h̄/2 . 9
σz 9 & −h̄/2 G & -
h̄/2 = 1 .
-
9 & 1 −1
-
-
9 . -
-
2 1 ;<I;<2 8 -
9
9 | + − . | − + 2 1
1 1 % ,
1 12 8 -
6
+
2 & , 12
⎛
1
1 ⎜
⎜
|ψ − = √ (| + − − | − +) = √ ⎜
2
2⎝
0
1
−1
0
⎞
⎟
⎟
⎟.
⎠
;6<
;6< 9
1& 1 -
9 E2 σz 9 9
! % 2 ! 11 1 h̄/2 σz 9
2 1 H2 -
11 1 ; <
1& 2 -
1 , &
2 n ;1 , &: |n + n+2 |n + n−2 |n − n+ . |n − n−<2 1 1
|ψ − = √ (|n + n− − |n − n+) .
;4<
2
2 1& & z 1C -
n
1& -
2 & E 9 2 9 9
2 : -
9 +2 z 2 9
−h̄/22 9
h̄/22 . x y
σn = nx σx + ny σy + nz σz n = (sin θ cos φ, sin θ sin φ, cos θ) #
"
σx =
σy =
σz =
0 1
1 0
0 −i
i 0
1 0
0 −1
;"<
,
,
;'<
,
; )<
$ %
4
?@ -
;6< ! 1 1
;2 -
&2 <
A!
9 ! D
B
&
A A # B )
θAB AB − cos θAB h̄/2 = 1
' #
( # !"!
!""
#
A a B b ∈ {−1, 1}
#
AB
AB
*
! −2 ≤ AB + Ab + aB − ab ≤ 2.
;
<
! A+a . A−a= . −2 2 2 B(A+a)+b(A−a)
& −2 2 $
-
A2 a2 B . b ,
1&1
,9 . & & E &
. -
&
1 -
G ; <
$
-
,9 9
9 H 2 . -
A . a 9 9
2 -
B . b 9
9
$
2 2 -
9
. E + 9 -
-
1 9
J
;2 & 5 &9< 1 9
2 A2 a2 B . b &E ! A . a2 B . b2 1
9 -
A . B 2 -
A . b2 -
a . B 2 . -
a
. b ;& E
< $
-
& 1&1 "
$ +
&' ,-, A a ! . /' B b !
. 2 1C 1 -
& ; < -
&+,#+
?@ ;!#$#< ?@ !#$#
$! !""
$ 12 ! -
A . B 9 -
, θAB 2 & AB − cos θAB 2
AB + Ab + aB − ab = − cos θAB − cos θAb − cos θaB + cos θab . ; <
√
cos θAB = cos θAb = cos θaB = − cos θab = 2/22 +
A = σx ,
; <
a = σy ,
; /<
√
B = −(σx + σy )/ 2,
√
b = (−σx + σy )/ 2,
√
1 2 22 -
. -
2
: ; <
; <
; 6<
! & !#$#
!%
. 1&12 & -
! & !#$#2 A
&
'
!#$# & -
!D
!( ?6@ -
√
√
−2 2 ≤ ABψ + Abψ + aBψ − abψ ≤ 2 2,
; 4<
ψ 2 A2 B 2 a . b . 1&1 C 2 & !#$# -
1
&
2 & 1 &
!
!
!"
# A a B b ∈ [0, 1]
! -
−1 ≤ AB − Ab + aB + ab − a − B ≤ 0.
; "<
C ; "< &1 A2 a2 B . b ;
E+ <2 C 1 1 &1 & 2 E $ A = a = 02
C & −B 2 -
& −1 0
$ A = a = 12 -
B − 12 -
& −1 0 $ A = 1 . a = 02
-
−b2 -
& −1 0 $ A = 0 . a = 12 -
b − 12 -
& −1 0
# 1 &1 AB → P (A = rA , B = rB )2 .
A → P (A = rA ) 2 2 1
−1 ≤ P (A = rA , B = rB ) − P (A = rA , b = rb )
+P (a = ra , B = rB ) + P (a = ra , b = rb )
−P (a = ra ) − P (B = rB ) ≤ 0,
; '<
P (A = rA , B = rB ) 11 2 A 1 9
. B 1 9
2 12 &2 rA . rB ; −1 1<2 . P (A = rA ) 11
2 A 1 9
2 1 rA ; '< &+ ?4@ ;!#<
)
&
!" !""
&+ :
; < & !# -
A2
a2 B 2 b ∈ [0, 1] & !#$#2 12
-
A2 a2 B 2 b ∈ {−1, 1} !#2 A2 1
!#$# .
&2 −1 12 E 2 !#
-
&1 2 %
2 !# -
&1 ;< !#$# 2 !# -
1 8 & -
A = rA 2 -
N (rA ) 8 N (rA , rB ) -
. 11 9 .
AB =
rA r B
{N (rA , rB ) − N (rA , −rB )
N
−N (−rA , rB ) + N (−rA , −rB )},
P (A = rA ) = N (rA ) /N,
P (A = rA , B = rB ) = N (rA , rB ) /N,
;)<
; <
;<
N 8 9
,
1+ E -
N -
!#$# & -
9
9 &
+
; < $ 12 1 &1 -
9
& +
. -
, !#$#2 -
1
, ! G,
2 !# 1 -
N 5 2 &C 9 , ; '<2 P (a = ra )+P (B = rB ) 1 1 . & P (a = ra ) + P (B = rB ) & 2
P (rA , rB ) − P (rA , rb ) + P (ra , rB ) + P (ra , rb )
≤ 1,
P (ra ) − P (rB )
;<
. -
& 8 N .
&+ '/ −1 ≤ P (A = rA , B = rB ) − P (A = rA , b = rb )
−P (a = −ra , B = rB ) − P (a = ra , b = −rb ) ≤ 0.
;/<
'
!" ( !""
1
11 -
!# . & -
!#$# & :
1
P (A = rA , B = rB ) = (1 + rA A + rB B + rA rB AB) ,
;<
4
1&1 2 + A2 -
1&1 A 1 9
B ( $
-
2 -
C 1 9
2 2 +2 A
.
&+ (0 &++
''
$)* !"
1&) 2 &+ (0 2 √
2 2
&++ ''3 1&) ) 2 &++ '' 3 1&) 2 &+
(03
#$% ! & '
'"' 012 # . 3 ?"@ ;0#3< 1
& 1 ! . ,
E %&
?'@ 9 0#3 :
;K< . 92 2 !#$# . !# ; 12 11<
;KK< & -
2 2 ) 4+5
> 0#3 -
1 2 +2
1
|GHZ = √ (|+ + + − |− − −) .
2
;6<
$ X1 σx 9 9
2 Y2 σy 9 9
2 Y3 σy 9
9
2 . 1+ -
h̄/2L 2 -
:
+
X1 Y2 Y3 = 1,
;4<
Y1 X2 Y3 = 1,
;"<
Y1 Y2 X3 = 1,
;'<
X1 X2 X3 = −1.
;)<
& (%678
> ;4<;)< , 1 -
X1 2 Y1 2 X2 2 Y2 2 X3 . Y3 2 -
-
5 1 9
$ -
9
E
+ 9 -
-
1 ,
2 -
1 9
. -
2 +2 X3 2 1
9
* E
$ 12 1C -
1 -
. 1
: -
X1 2 Y1 2 X2 2 Y2 2 X3
. Y3 & & E −1 1 & 2 +2
X1 9 . $ 12 X1
Y1
Analizador de
espines nº 1
Analizador de
espines nº 3
Analizador de
espines nº 2
Fuente de
estados GHZ
Y3
X2
Y2
X3
$ 0
&' 1,2 Xi Yi ! . i
/
1 & & -
& &2 -
19 1 5-
2
-
19 −1 G 0#32 ? )@ & ; < -
1 -
12 1& -
;6< & -
O = σx1 σy2 σy3 + σy1 σx2 σy3 + σy1 σy2 σx3 − σx1 σx2 σx3 ,
; <
& / & 9
-
1 &
2 -
1&1 & E
-
9:
−2 ≤ A1 B2 B3 + B1 A2 B3 + B1 B2 A3 − A1 A2 A3 ≤ 2,
;<
Ai . Bi ;i = 1, 2, 3< 1&1 C 1 9
i
,
1&) ) 2 9
7(3
F -
& . 9
. -
1 -
!#$# -
1 2 -
. -
1 E -
-
12 9
&+ -
!#$# , -
&E2 51 2 ! 0#3 ? @
!&
''2 #. ; 11< & -
1 5 ? @ #. 11 1 #. -
-
-
1 -
5 52 |η2 1&1
; 92 +< A . a 9
. 1&1 B . b 9
2 -
1 ,:
|η = c++ |A + B+ + c+− |A + B−
+c−+ |A − B+ + c−− |A − B−
;<
= d++ |A + b+ + d−+ |A − b+ + d−− |A − b−
;/<
= f++ |a + B+ + f+− |a + B− + f−− |a − B−
;<
= g+− |a + b− + g−+ |a − b+ + g−− |a − b−,
;6<
E ci 2 di 2 fi . gi % -
|η :
Pη (A = +1, B = +1) = |c++ |2 ,
;4<
Pη ( b = +1 | A = +1) = 1,
;"<
Pη ( a = +1 | B = +1) = 1,
;'<
Pη (a = +1, b = +1) = 0.
;/)<
! -
. A ;1 9
< . B ;1 < . E+C .
1 1 +1 2 8 ;4< K -
1C
1 9
1&1 b 1&1
B $8 ;"< 19 2 5 b = +1 % 2 -
9
1 A = +1 1 5 . 1 9
; -
9
E +< -
& b +1 2 8 2 9
b = +1 2 ;'< -
9
a = +1 2 1C a 1 9
. b 1 9
2 19 1 a = +1 . b = +1
$ 12 ;/)< 2 5 ; +#<
* E
/
6
$ 3
&' ,
σx ⊗ σx : 2
&
(
σz ⊗ σx σx ⊗ σz #
σz ⊗ σz 1
|η = √ (| + + + | + − + | − +).
3
|η ;/ <
-
Pη (z1 = −1, z2 = −1) = 0,
;/<
Pη (z1 = −1 | x2 = −1) = 1,
;/<
Pη (z2 = −1 | x1 = −1) = 1,
;//<
Pη (x1 = −1, x2 = −1) > 0.
;/<
!& & !
M & -
!# ; '< 9 1 C ;/< , C 11 -
11 #. & ;/< rA = rB = ra = −rb = 12 9 1
−1 ≤ P (A = 1, B = 1) − P (A = 1, b = −1)
−P (a = −1, B = 1) − P (a = 1, b = 1) ≤ 0,
4
;/6<
11 ;4<I;/)< -
2 -
#. 5 2 2 -
E ,2 #. & ;. .
2 -
< !#
)
*
0#3 ,2 &
5 9
#. & 5 9
2 2
5 ; <2 -
-
-
& :
;< A$ #. 5 D
;1< A$ ,D
9 1 ? 2 /@ % 2 , + 9
. 9 ; z x< 1C 1&1
9
2 + σz1 ⊗σz2 ;
. Z1 Z2 <
$
-
9
. . -
1C 9
. / $
1C
-
1& 1 9
. 2
-
1& E + 1 9
. / * E
2 , -
,
1 -
19 0#3 . -
1
& E2 −1 12 1&1 ; +
- √ 4%+5 ' 6! ' 7089 [( 5 − 1)/2]5 ≈ 0,09
"
$ :
&' ( Xi Zi ! . i Z1 Z2 ! . % + '
, 9
. 2 . + , 9
. /< -
& :
Z1 = −Z3 ,
;/4<
X1 = −X3 ,
;/"<
Z2 = −Z4 ,
;/'<
X2 = −X4 ,
;)<
Z1 Z2 = Z3 · Z4 ,
; <
X 1 X 2 = X3 · X4 ,
;<
Z1 · X2 = Z3 X4 ,
;<
X1 · Z2 = X3 Z4 ,
;/<
Z1 Z2 · X1 X2 = −Z3 X4 · X3 Z4 ,
;<
. : 1&1 . 2 5-
1 -
−1 2 -
, 2 & 1 -
1 G 2 &52 & &
? /@
+
, 1 1 & 5 -
12 W . 0#3 ? @
G -
1 -
12
1 1 & !#$# . -
!( ? 6@
σn =
cos θ
e−iφ sin θ
iφ
e sin θ − cos θ
)
,
;6<
# |n+ =
|n− =
e−iφ/2 cos 2θ
eiφ/2 sin 2θ
;4<
,
−e−iφ/2 sin 2θ
eiφ/2 cos 2θ
;"<
.
+ |n + n+ |n + n− |n − n+ |n − n−
n + |
n − |
⊗
n + |
n − |
.
⎛
1 ⎜
⎜
·√ ⎜
2⎝
$ 0
1
−1
0
⎞
⎛
⎟
1 ⎜
⎟
⎜
⎟ = √ ⎜
⎠
2⎝
z
0
1
−1
0
⎞
⎟
⎟
⎟ .
⎠
;'<
n
* * 2 " A = σz ,
;6)<
B = sin θAB σx + cos θAB σz .
;6 <
:
AB = A ⊗ B
=
⎛
⎜
⎜
⎝
= ⎜
: 1 0
0 −1
⊗
cos θAB
sin θAB
sin θAB − cos θAB
cos θAB
sin θAB
0
0
sin θAB − cos θAB
0
0
0
0
− cos θAB − sin θAB
0
0
− sin θAB cos θAB
AB
ψ
;6<
;6<
⎞
⎟
⎟
⎟.
⎠
;6/<
ABψ = ψ|AB|ψ
;6<
= − cos θAB .
;66<
;
, '/-
P (a = ra , B = rB ) − P (B = rB ) = −P (a = −ra , B = rB ) ,
;64<
P (a = ra , b = rb ) − P (a = ra ) = −P (a = ra , b = −rb ) .
;6"<
; ' θ = φ = 0 |+ |− " 7+9<7:9
−1 ≤
# (< (0 1
(−2 + rA rB AB − rA rb Ab + ra rB aB + ra rb ab) ≤ 0.
4
;6'<
9
0 # ( −2 ≤ rA rB AB − rA rb Ab + ra rB aB + ra rb ab ≤ 2,
) &++ :
rB = ra = −rb = 1
;4)<
rA =
−2 ≤ AB + Ab + aB − ab ≤ 2,
;4 <
&++ ''
& ) &+ (0 # &++ '' 2
lCH
√
= (lCHSH −2)/4 =
( 2 − 1)/2
2 −1
2
1
4
''
'' = , 2 &+ (0 1/2
+
)
:
|GHZ ($ ) σz1 σz2 # σz3 σx1 σy2 # σy3 2 (% -
1
(|x + y + y+ + |x + y − y− + |x − y + y−
2
+ |x − y − y+),
1
=
(|y + x + y+ + |y + x − y− + |y − x + y−
2
+ |y − x − y+),
1
(|y + y + x+ + |y + y − x− + |y − y + x−
=
2
+ |y − y − x+),
1
(|x + x + x− + |x + x − x+ + |x − x + x+
=
2
+ |x − x − x−).
|GHZ =
,
=
7( ) −4
;4<
;4/<
;4<
−1
#
1
2 9 ) ;4<
4
= ) 2 7( 4+5 ($ # |η =
=
=
1
√ (2 |z + x+ + |z − x+ − |z − x−),
6
1
√ (2 |x + z+ + |x + z− − |x − z−),
6
1
√ (3 |x + x+ − |x + x− − |x − x+ − |x − x−).
2 3
;46<
;44<
;4"<
> 0(60< 0' %$6%?
: %?
Pη (x1 = −1, x2 = −1) =
1
.
12
;4'<
? @ G2 N -
.2 , @A BC 2 /
. ' O2 (
N.2 > D,,
2
2 ) ' ; '")<
?@ G 2 N. . F 2 ! -
, . . 1 D2 =,# D ,2
4444") ; '<
?@ O $ 2 7 N. 2
')) ; '6/<
=,#
2 2
?/@ % 2 E
,#2 #2 P !Q2 FP O.2
' = !12 2 '/= %&2 F
& MN2 '"'2 6 /6
?@ O R !
2 G #2 G $. . G #2 &1 2 =,# D >
2 2
"")""/ ; '6'<
?6@ $ !(2 B
5 , ( -
.2
9
, =,# 2 2 ' )) ; '")<
>
?4@ O R !
. G #2 -
, 1+&
2 =,# D . 2 2 6 ; '4/<
?"@ % 012 G # . G 32 0 1. (
2 S, ;<2 ;F , ,# , ? , PN ;0
>&.2
'""<@2 S
P G2 %2 #2 '"'2 6'4
?'@ F %
#
2 T( P P , .D2
2 62 ' ; '')<
=,#
? )@ F % 2 -
, . 2 =,# D >
+2 2 "" "/)
; '')<
? @ OT 2 % P2 %2 # T,
. G 32
, -
. 01
#3 2 @
2 646'2 ' ;)))<
? @ #.2 F. , P P
-
, 2 =,# D >
, 2 2 66 66" ; ''<
/
? @ G !12 ( P
-
P
11
, P 1&2 =,# D >
+2 )2 ' ' / ;)) <
? /@ G !12 UG &
1. , P 1&(2 =,#
D >
,2 2 ) )/) ;)) <
? @ G !12 ( P P
-
, -
1 01#3 W 2 =,# D G +2 2
) )" ;))<
? 6@ G !12 * ( -
. 1. !(( 1
2 =,#
D >
2 62 )6)/) ;))<