! " #$ % & ' ( !" ! # $ % $ & ' ( ) # * #+ , + $ " - + & # ' & ! ! !#$# %& !#$# ! & !#$# !( !# %& !# *+ !# , !#$# - & !# . !#$# / & !# / 6 012 # . 3 #. #. . !# 5 7 " " " ' ' ) ) 4 " ) $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 = −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