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Medidas de entrelazamiento cuántico de sistemas tripartitos de qubits y sus aplicaciones experimentales

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(1)                                            .

(2)       . .    .                                                                .                                  .    .                             ! "             |GHZ    |W       .                   .    .                                 .     .          .          .       $     %             %              %           &                        '( )        .    .   .          .          .          .          .          .          .          .          .          .          .          .          .          .          .             %        .      .      .      .      .      .      .      .      .      .      .  . .    .  # . .     . .

(3)                                                                                                                                              * +, -  , .** $/0 1.

(4)        $  ,   *         %          %      *              .  23         )         4       % $ ,         %      3 ,        $  5 23  *            4          -   /     3    - ,2 6,6/3      $       4          (          $       5      3  ,    $                      3     ,  GHZ $ W 3    % 7   . 8 3    ,       %     9      %     3               -2  7       / $       3    23         :    *   7             %    3          $ )  )  3    ;, $   3         GHZ  < 7      $  2                 . .

(5)                          %   * +,   3 %     " *=> , -  %/3 72   ?* * 4        *@ 0 1 .            )     2   (  3 .> $ &   3    7    ( .& 01      %  )       %   4                  3    *        3         %            <   ,    (         "  7        A         " 0 1 - ( B  "   ,3     $ (3    *  $ "   /    7  0C $ 1C   %     % |0 $ |1    < 3                3  ( B |ψ = α|0 + β|1,. -  /.  |α|2              |03 |β|2         |1 $      |α|2 + |β|2 = 1 <8       %    * 9    )   3  % %     %        % %     .   8    *        $ .      A               -   %/ |ψ       ρ3    % 7     .

(6)  . .     .  %  B. ρ = |ψψ|. -  /. $     tr(ρ2 ) = 1.. $. tr(ρ) = 1. - /. ' %                                    3   3          . ( 3     %     n  3        |ψi 3                n  |ψ = |ψi , -  / ⊗.             . (    |ψ = α|00 + β|11. -  /. %                  .      |ψ. = α|01 + β|11. - /. = (α|0 + β|1)|1 = |ψ1 |ψ2.  %            3       %   4           *$             %   . ( 3  |ψ               $ k      n         %                |ψ = |ψ  . k . |ψi .. - /. ⊗. A (     |ψ. = α|0011 + β|1111 = (α|01 |02 + β|11 |12 )|13 |14 = |ψ  . 4  ⊗i=3. |ψi .. - /. - /.

(7)  . .   .     . . . :              ρ   3        3         4       0 1 :             B $. tr(ρ) = 1. tr(ρ2 ) ≤ 1;. -  #/. $                           "  B  ρ= pi |ψi ψi |. -  / i.   ,     .  0 1  ρ= wA ρA ⊗ ρA ,. - . /. A.  ρA $ ρA            $         %  .  A. wA = 13.

(8)             .     .  . :         3     D D  013                   |ψ      < $ E  7  = σy ⊗ σy |ψ ∗ , |ψ A B.  |ψ ∗     ( ( ,  |ψ $ σy    .    7     B  C = |ψ|ψ|,.     *      . ( 3       .   $   . -  /    .   4 -  /. . |ψ = α|01 + β|10,. - /.  = β ∗ |01 + α∗ |10, |ψ. -  /. C = |αβ ∗ + βα∗ |.. -  /. .   7 3  α $ β  "     0 $ 1 $ |α|2 + |β|2 = 13         √   "   0 $ 1  2     α = x $ β = 1 − x2   x     0 ≤ x ≤ 13  B  |ψ = x|01 + 1 − x2 |10, - / .

(9)     .     . . C 1 0.8 0.6 0.4 0.2. 0.2. 0.4. 0.6. 0.8. 1. x. F,   B       x :        0   x = 0 $ x = 1B      G    %)  1   x = √12 B   %    . $.  = |ψ. . . 1 − x2 |01 + x|10,.  C = |2x 1 − x2 |,. - / - /.     , %7   7,   .   . . .               % |ψ  7    ρAB = |ψψ|. - /. 57  ρAB   01B ρAB = σyA σyB ρ∗AB σyA σyB ,. -  #/.  ρ∗AB    ( ( ,  ρAB   λ1 , λ2 , λ3 $ λ4  "          ρAB ρAB H    01    <E B     τAB = máx[ λ1 − λ2 − λ3 − λ4 ; 0]2 . -  / |ψ     3    ρAB     7 " 5    3  ,  )        ρAB    )       ,         .   ρAB ρ̃AB.

(10)     .     . .    "   " λ1                 B τAB = λ1 . -  /   ,   ,    3    %               -     /B C2. ⇒ C2.  2 = |ψ|ψ|  ψ|ψ  = ψ|ψ. -  /.  ψ|ψ|)  = tr(|ψ|ψ  ψ|)  = tr(|ψψ||ψ = tr(ρAB ρAB ) = λ1. -  /. -  -. = τAB.    . / / / / /. <         (            B  ρAB = x2 |0101| + (1 − x2 )|1010| + x 1 − x2 (|0110| + |1001|). -  #/    ρAB B  ρAB = x2 |1010| + (1 − x2 )|0101| + x 1 − x2 (|0110| + |1001|), - . /.    ρAB ρAB. =. 2x2 (1 − x2 )(|1010| + |0101|) + 2x3 + 2x(1 − x2 )3/2 |1001|,. . 1 − x2 |0110| + - . /.   "   B λ1 = 4x2 (1 − x2 ) λ2 = λ3 = λ4 = 0.. -  / -  /.   $         B τAB = 4x2 (1 − x2 ).. -  /. ! " 7,   . .  . .          <E3      01 01  4          <  E .                             .

(11)     .     . #. F,   B       x         0   x = 0 $ x = 1B      G    %)  1   x = √12 B   %    . :    3     I J 3     "            ,       -  4        /         -       / K       3         3             $        %  H                )  "           <E              :                # :        <       E     #B ρA ρA. 5 .    ρB ρB. = trB (ρAB ) = 0|B ρAB |0B + 1|B ρAB |1B = x2 |00| + (1 − x2 )|11|.. 3 .     E B. = trA (ρAB ) = 0|A ρAB |0A + 1|A ρAB |1A =. -  / -  /. 2. 2. (1 − x )|00| + x |11|.. -  / -  /. :       <  7  B hA. = −tr(ρA ln2 (ρA )). hA hA. = −x2 0| ln2 (ρA )|0 − (1 − x2 )1| ln2 (ρA )|1 = −x2 ln2 (x2 ) − (1 − x2 ) ln2 (1 − x2 ).. - #/ -  / -  /.

(12)     .     . h 1 0.8 0.6 0.4 0.2 x 0.2. 0.4. 0.6. 0.8. 1. F,  B         x :          0   x = 0 $ x = 1B      G    %)  1   x = √12 B   %    . 5 , .        h = hA = hB .. - /. ! " 7,  . .     .   . .        < $ E            &"   x         )  ,      C B  √ 1 − 1 − C2 x = − -  / 2  √ 1 + 1 − C2 x = − -  / 2  √ 1 − 1 − C2 x = - / 2  √ 1 + 1 − C2 . x = - / 2 .      3            "   0 $ 1   0 ≤ C ≤ 1 5                 .

(13)     .     . F,   B            τAB    4          0   τAB = 0B        G  " %)  1   τAB = 1B           3     7            "    x       01B.  ξ   - /. ξ = h(x),.  .  x=.   .  ξ=h. ξ. 1+. √. 1+. 1 − τAB , 2. √. 1 − τAB 2. - /. √ 1 + 1 − τAB 1 − τAB ln2 − 2 2 √ √ 1 + 1 − τAB 1 + 1 − τAB ln2 1 − − 1− 2 2. = −. 1+. -  #/. ,. √. .. - . /. I2 7,   . .     .         3   "               B      x = 0 $ x = 1 $   %)  " 1   x = √12 -"   $ 7,   /.

(14)     .     . '              .      0 0 0 0. .     1 1 1 1.     B &                 B  3  3     $      4 . F,   B      , %7    - (/3   -" / $     - / :        %)   x = √12 $          3    x = 0 $ x = 1. :                  E3     ( B |ψ =. |01 + |10 √ . 2. - . /. A     2            3   x = sin θ $ 0 ≤ θ ≤ 2π   0 ≤ x ≤ 1B |ψ = sin θ|01 + cos θ|10.. :.   )  *     "  4 . -  /. B. B -    / C = | sin(2θ)|.. -  /.

(15)     .     . F,  B L %7     - (/3   -" / $     - /     θ   0 ≤ θ ≤ π2  :      2    θ = π4 3    %)    . 4  B -     / τAB = sin2 (2θ).. -  /. 4    B -     / τAB = − sin2 θ ln2 (sin2 θ) − cos2 θ ln2 (cos2 θ).. -  /.   , 7   " ) 3 7,  3            2    θ = π4 3          √ %)     |ψ = |01+|10  2 .    , 7 *      3 7,  3     7,      *(     7, 3      *(  4 √  4            |ψ = ±|01±|10 2  *(   $    x = y = 0 -       B |ψ = ±|01  |ψ = ±|10/3       *  %)  -    *(/    ,       :     *(             4 "     E3     3 β01 $ β11 0 1B 4   r = 1 $ θ =. π 4. β01 =. |01 + |10 √ , 2. -  /.

(16)     .      y 0.75. 0.5. 0.25. x -0.75. -0.5. -0.25. 0.25. 0.5. 0.75. -0.25. -0.5. -0.75. F,  B L %7     - (/3   -" / $     - /      0 ≤ θ ≤ 2π 2       *(       . 4   r = 1 $ θ =. 3π 4. |01 − |10 √ . -  / 2 √ √ :     E3 |β00  = (|00 + |11)/ 2 $ |β10  = (|00 − |11)/ 23                        β11 =.   .    .            < 3E $          |ψ     7  ,    01B  = σy ⊗ σy ⊗ σy |ψ ∗ , |ψ A B C ρABC = σyA σyB σyC ρ∗ABC σyA σyB σyC ,. -  / -  #/.          H   3    ρABC $      7        3  $ # "  .         *  , B .   3      <E3 E $ <B ρAB. = trC (ρABC ). - . /.

(17)     .     . ρAB ρBC. = 0|C ρABC |0C + 1|C ρABC |1C = trA (ρABC ). ρBC ρAC. = 0|A ρABC |0A + 1|A ρABC |1A = trB (ρABC ). ρAC. = 0|B ρABC |0B + 1|B ρABC |1B ..  -.  /  /  /  /  /. -.  /  /  / #/  /  / /  /  /. F         <3 E $   *B ρA ρA. = trB (ρAB ) = trC (ρAC ) = 0|B ρAB |0B + 1|B ρAB |1B. ρA ρB. = 0|C ρAC |0C + 1|C ρAC |1C = trA (ρAB ) = trC (ρBC ). ρB ρB. = 0|A ρAB |0A + 1|A ρAB |1A = 0|C ρBC |0C + 1|C ρBC |1C. ρC ρC ρC. = trB (ρBC ) = trA (ρAC ) = 0|B ρBC |0B + 1|B ρBC |1B = 0|A ρAC |0A + 1|A ρAC |1A .. <*              |GHZ $ |W .  . .    .  |GHZ. A  |GHZ                  %  B 1 |GHZ = √ (|000 + |111). - / 2               |GHZ     |GHZ , B |ψGHZ  = α|000 + β|111,. - /.  α $ β  8    (3 |α|2 + |β|2 = 13 |α|2           2  |000 $ |β|2       2  |111 <            B |ψGHZ  = β|000 + α|111,.     . - /.  B. C = |αβ ∗ + βα∗ |. - / √   3     α = x $ β = 1 − x2   x  $ 0 ≤ x ≤ 1  B  C = 2x 1 − x2 . - #/.

(18)     .     . . C 1 0.8 0.6 0.4 0.2. 0.2. 0.4. 0.6. 0.8. 1. x. F,  B  √      x             |ψGHZ  = x2 |000 + 1 − x2 |111 :     "   0   x = 0 $ x = 13        G    %)  " 1   x = √12 3     %      "    ,   √             |ψ = x2 |01 + 1 − x2 |10. :       )              7,  <*      .             -  /3        , %7     .    .           4    <3 E $ 3      < $ E3 E $  $ < $  K            $   3   $                B ρAB = |α|2 |0000| + |β|2 |1111| ρBC = |α|2 |0000| + |β|2 |1111| ρAC = |α|2 |0000| + |β|2 |1111|.. <*      3 # $          B ρA = |α|2 |00| + |β|2 |11| ρB = |α|2 |00| + |β|2 |11| ρC = |α|2 |00| + |β|2 |11|.. -  / -  / - /    -  / -  / - /.

(19)     .     . . h 1 0.8 0.6 0.4 0.2 x 0.2. 0.4. 0.6. 0.8. 1. F,  B     √     x             |ψGHZ  = x2 |000 + 1 − x2 |111 :       "   0   x = 0 $ x = 13        G    %)  " 1   x = √12 3     %      "   √,                 |ψ = x2 |01 + 1 − x2 |10.           |ψGHZ    ,         "   α $ β B. $. ρAB = ρBC = ρAC ,. - /. ρA = ρB = ρA .. - /. 5               |GHZ   B. . h = hA = hB = h A .. - /. h = −|α|2 ln2 (|α|2 ) − (|β|2 ) ln2 (|β|2 ),. - #/.     ,   "     √                α = x $ β = 1 − x2 B h = −x2 ln2 (x2 ) − (1 − x2 ) ln2 (1 − x2 ).. -  /. I 7,   5                       B τAB = τBC = τAC , -  /.

(20)     .     . . 7             *     ρAB     <EB. .  . ρAB = |β|2 |0000| + |α|2 |1111|,. - /. ρAB ρAB = |α|2 |β|2 (|0000| + |1111|),. -  /.    "   B λ1 = λ2 = |α|2 |β|2 λ3 = λ4 = 0.. -  / - /. .       <E B     τAB = máx[ λ1 − λ2 − λ3 − λ4 ; 0]2 τAB = 0.. - / - /.                %  3   3 <  %    E3 <  %     $ E  %     .       4 3 ,8 "       <3 E $        7  4    01B τABC. = τA(BC) − τAB − τAC. τABC τABC. = τB(AC) − τAB − τBC = τC(AB) − τAC − τBC .. - / - #/ -  /.         )    " .        τAB = τBC = τAC H 7  τA(BC) 3 τA(BC) $ τA(BC) 01B -  / - / -  /. τA(BC) = 4 det(ρA ) τB(AC) = 4 det(ρB ) τC(AB) = 4 det(ρC ).. .      ρA = ρB = ρC 3  det(ρA ) = det(ρB ) = det(ρC ) $.   . τA(BC) = τB(AC) = τC(AB) ,. -  /. det(ρA ) = |α|2 |β|2 ,. - /. τA(BC) = τB(AC) = τC(AB) = 4|α|2 |β|2 .. - /.   $      )     4  - 3 #3 $  /   " .         $ 3   3           B τABC τABC. - /. = τA(BC) =. 2. 2. 4|α| |β| .. - /.

(21)     .     . #. F,   #B  √    x             |ψGHZ  = x2 |000 + 1 − x2 |111      "   0   x = 0 $ x = 13        G    %)  " 1   x = √12 3     %      "    ,   √               |ψ = x2 |01+ 1 − x2 |10 √     α = x $ β = 1 − x2        4     )  $    , %7               -" 7,   #/B τABC = 4x2 (1 − x2 ).. K            |GHZ     .   . . -  ##/ %)      4.  |W . A  |W                   %  BB 1 |W  = √ (|001 + |010 + |100). -  # / 3 .             ,  |W B |ψW  = α|001 + β|010 + γ|100,. -  # /.  α3 β $ γ  8    (        4  |α|2 + |β|2 + |γ|2 = 1 |α|2              |0013 |β|2         |010 $ |γ|2           |100.

(22)      ,  .     .           B |ψW  = α|110 + β|101 + γ|011,. $     . -  #/.  B C = 0.. -  # /.   |ψW     3   %  3        3        2             ,               .    .            )             3              :           <E3 < $ E  " B ρAB. = |α|2 |0000| + |β|2 |0101| + |γ|2 |1010| + + βγ ∗ |0110| + β ∗ γ|1001|. ρBC. = |α|2 |0101| + |β|2 |1010| + |γ|2 |0000| + + αβ ∗ |0110| + α∗ β|1001|. ρAC. -  # /. = |α|2 |0101| + |β|2 |0000| + |γ|2 |1010| + + αγ ∗ |0110| + α∗ γ|1001|.. 5            B. -  #/. -  #/.            . ρA ρB. = =. (|α|2 + |β|2 )|00| + |γ|2 |11| (|α|2 + |γ|2 )|00| + |β|2 |11|. ρC. =. (|β|2 + |γ|2 )|00| + |α|2 |11|.. -  #/ -  #/ -  #/. :                   3  ,              .    *$  23    3           K 2        α = β = γ = √13                3  ,                4  τABC       .     τA(BC) 3  ,  τB(AC) $ 7   τC(AB)   %   4   τABC = τA(BC) − τAB − τAC B •τA(BC). =. 4 det(ρA ). =. 4|γ|2 (|α|2 + |β|2 ).. - . /.

(23)     .     . I     ρAB ρAB B 4|β|2 |γ|2. λ1. =. λ2. = λ3 = λ4 = 0     = máx[ λ1 − λ2 − λ3 − λ4 ; 0]2 = 4|β|2 |γ|2 .. •τAB. - . /. I     ρAC ρAC B 4|α|2 |γ|2. λ1. =. λ2. = λ3 = λ4 = 0     = máx[ λ1 − λ2 − λ3 − λ4 ; 0]2 = 4|α|2 |γ|2 ,. •τAC. . τABC = 0.. -  / - . /. - . /.  %   4   τABC = τB(AC) − τAB − τBC B •τB(AC). =. 4 det(ρB ). =. 4|β|2 (|α|2 + |γ|2 ).. I     ρBC ρBC B λ1 λ2 •τBC. =. 4|α|2 |γ|2. = λ3 = λ4 = 0     = máx[ λ1 − λ2 − λ3 − λ4 ; 0]2 =. 4|α|2 |β|2 .. A    . -  /.  B τABC = 0.. - . /.  %   4   τABC = τC(AB) − τAC − τBC B •τC(AB). A    . =. 4 det(ρC ). =. 4|α|2 (|β|2 + |γ|2 ). $ . -  /.   B. τABC = 0.. - . /.

(24)     .     . . :           4   ,  $  3     3   2  )   3         %              ,3     "        .   3                 3  (  <    E3  E  <    <E , 3 )          3 <  E3 <   $ E   .          "   )    ,  |W     2 B |ψW  = sin θ cos φ|001 + sin θ sin φ|010 + cos θ|100.. 5     4  -      /B τAB. τAC. τBC. = =. 4 sin2 θ sin2 φ cos2 θ sin2 (2θ) sin2 φ. =. 4 sin2 θ cos2 φ cos2 θ. = = =. - . #/. 3   $. - . /. sin2 (2θ) cos2 (φ). - . /. 4 sin2 θ cos2 φ sin2 θ sin2 φ sin4 θ sin2 (2φ).. - . /.         $ , %7 -7,   /   4    "   τAB $ τAC      3     φ  φ + π2            3     " 3     φ  φ + π2               3      τAB    π2   ( X       τAC  "   -" 7,   -/ $  -//   7,                   4       $ 7  x = y = z = 0 -      4 /         -|ψW  = ±|001  |ψW  = ±|010  |ψW  = ±|100/ $          *  %)  - )4       /3                 3   )     7,   -/          <E3     7,   -/     E      < $     -/    <  %      E     3   φ = 0 E     < $    7,       -XZ /   7,   -/H   φ = π2           <E $    7,       -Y Z /   7,   -/H  ,       7,   -/   7,         -XY /  <  %    E.

(25)     .        . . .     . .        .        .   .   . . . F,   B &       4       ,  |W     τAB  π2    ( x   τAC 3         τAB     π2 τAC  : )     7,     %)         3      "     .

(26)     .     .    *            -  2,1113 2,115 $ 2,118/B =. 4 cos2 θ(sin2 θ cos2 φ + cos2 θ + sin2 φ). =. sin2 (2θ). - . /. τB(AC). =. 4 sin2 θ sin2 φ(sin2 θ cos2 φ + cos2 θ). - . /. τC(AB). =. 4 sin2 θ cos2 φ(sin2 θ sin2 φ + cos2 θ).. - . /. τA(BC). 5       4 3         $ , %7 -7,   /                τB(AC) $ τC(AB)      3     φ  φ + π2          3  "     3      τB(AC)    π2    ( X       τC(AB)  "        -" 7,   -/ $  -// !         τA(BC)     2     ( Z 3         φ  ( " H             θ ( "  τB(AC)      2    ( Y H 7  τC(AB)     2    ( X .    *    B     "  φB .  φ = 0B |ψ. τAB τAC τB(AC). = sin θ|001 + cos θ|100 = (sin θ|01AC + cos θ|10AC )|0B. - . = τBC = 0 = sin2 (2θ). -  / -  /. =. /. -  #/. 0 2. τA(BC). = τC(AB) = sin (2θ). -   /. ⇒ τAC. = τA(BC) = τC(AB). -   /.

(27)     .           .     .  . .           .            . .  . . . . F,   B &                   ,  |W   τB(AC)    π2   ( X    τC(AB) 3         τB(AC)  τC(AB)    π 2.

(28)     .     . . .  φ = π2 B |ψ. τAC. =. sin θ|010 + cos θ|100. =. (sin θ|01AB + cos θ|10AB )|0C. -   / -   /. = τBC = 0 2. τAB. =. sin (2θ). τC(AB). =. 0. -  /. -  / 2. τA(BC). = τB(AC) = sin (2θ). -  /. ⇒ τAB. = τA(BC) = τB(AC). -  /. |ψ. -  /.     "  θB .  θ = 0B =. cos φ|100. = τAC = τBC = 0. - . #/. = τB(AC) = τC(AB) = 0. - . /. = cos φ|001 + sin φ|010 = (cos φ|01BC + sin φ|10BC )|0A. - . /. -  - . / /. - . /. τAB τA(BC). .  θ = π2 B |ψ. τAB. = τAC = 0 2. τBC. =. sin (2φ). τA(BC). =. 0 2. τB(AC). = τC(AB) = sin (2φ). -  /. ⇒ τBC. = τB(AC) = τC(AB). - . /.   (       " ,      4            "    |GHZ                        , B.

(29)     .     . . 0.6. 0.4. 0.2. -0.6. -0.4. -0.2. 0.2. 0.4. 0.6. -0.2. -0.4. -0.6. F,   B    4               3   φ = 0  φ = π2  θ = π2 3        |W   )     *(    %)         . 4        3     3     -4 3 4  $           /  ,    H 4   )           3                         3                      3  ,   4     H 4      3             E    3     θ = π2 3 " 7,   3 $     %) )       , 2  -7,  .   |GHZ.      4  φ = 0  φ = π2      /.         .  |GHZ.     ρIGHZ =. ) 7  B. x I + (1 − x)|GHZGHZ|, 8. - . /.   0 ≤ x ≤ 13   x = 1            3   x = 0       .

(30)     .     . .    |GHZ )  )   B ρIGHZ. =. x I+ 8 (1 − x) (|000000| + |111111| + + 2 + |000111| + |111000|),. - . /. - . #/. ρA = ρB = ρC , 1 -  ρA = (|00| + |11|). 2 = τAC = τBC $  "     ρAB ρAB B. /.    ρAB. $. ρAB = ρAC = ρBC , x 2−x (|0000| + |1111|) + (|0101| + |1010|) = 4 4.   . # τAB. λ1 λ3 ⇒ τAB. $  . 2 − x2 16 x2 = λ4 = 16 = 0; = λ2 =. /. τA(BC) = τB(AC) = τC(AB) B τA(BC) = 4 det ρA = 1..   4  B. τABC = 1..  4    |GHZ "  x |W . - . -  / - . /.           .   .  |W .     ρIW =. ) 7  B. x I + (1 − x)|W W |, 8.   0 ≤ x ≤ 13     x = 1         x = 0       |W  ρIW. =. - . /. 3. x I+ 8 1−x (|001001| + |010010| + |100100| + + 3 + |001010| + |010001| + |001100| + |100001| + + |010100| + |100010|).. -  /.

(31)     .     . #. 5        B ρAB = ρAC = ρBC , ρAB. =. x 4−x (|0000| + |0101| + |1010|) + |1111| + 12 4 1−x + (|0110| + |1001|) 3. $. - . /. - . /. ρA = ρB = ρC , 2+x 4−x |00| + |11|. 6 6 = τB(AC) = τC(AB) B. ρA =. 5 .    τA(BC). τA(BC) = 4 det ρA =. (4 − x)(2 + x) , 9. -  /.         $ "  89  1   0 ≤ x ≤ 1 .    τAB = τAC = τBC $     "     ρAB ρAB B λ1 λ2 λ4 ⇒ τAB. 1 (5x − 8)2 144 x(4 − x) = λ3 = 48 x2 = 16 2 = máx [λ1 − λ2 − λ3 − λ4 ; 0] . =. -  #/. .        #  "   0 ≤ x ≤ 0, 45 $ ,"   0, 45 ≤ x ≤ 13   B ⎧ √ 2 ⎨ 2−2x x(4−x) √ −   0 ≤ x ≤ 0, 45 3 2 3 τAB = -   / ⎩   0, 45 ≤ x ≤ 1 0 $ τABC =. ⎧ ⎨ ⎩. (4−x)(2+x) 9. −2. 2−2x 3. −. (4−x)(2+x) 9. √. x(4−x) √ 2 3. 2.   0 ≤ x ≤ 0, 45. -   /.   0, 45 ≤ x ≤ 1.  4   |W         $   -7,   -//     #       |W  $       4  -               |GHZ /.

(32)     .     . .   .  .       .  . . |W    . F,   B  , %7            |W   x = 0        |W  $ x = 1         .

(33)         |GHZ.     . . |W .  7  , .   B. ρW GHZ = x|GHZGHZ| + (1 − x)|W W |,. -  /.   0 ≤ x ≤ 13  x = 0       |GHZ $   x = 1       |W  :       ρAB = ρAC = ρBC $ ρA = ρB = ρC 3  τA(BC) = τB(AC) = τC(AB) $ τAB = τAC = τBC  5   ρAB. =. x 2+x |0000| + |1111| + 6 2 1−x + (|0101| + |1010| + |0110| + 3 + |1001|). -   /. $ ρA =. 2+x 4−x |00| + |11|. 6 6. -   /.     B. (4 − x)(2 + x) . 9 G     "     ρAB ρAB B ⎧ ⎨ λ1 = 49 (1 − x)2 x λ2 = λ3 = 12 (2 + x)   0 ≤ x ≤ 0, 51 ⎩ λ4 = 0 $ ⎧ x (2 + x) ⎨ λ1 = λ2 = 12 4 λ3 = 9 (1 − x)2   0, 51 ≤ x ≤ 1 ⎩ λ4 = 0 τA(BC) =. -  /. -  /. -  /. .    τAB = máx [λ1 − λ2 − λ3 − λ4 ; 0]2      $  B   2    0 ≤ x ≤ 0, 29 τAB = 23 (1 − x) − x 2+x 3 -  /   0,29 ≤ x ≤ 1 τAB = 0 $ ⎧  2  ⎨ 2 2+x τABC = (4−x)(2+x) − 2 (1 − x) −   0 ≤ x ≤ 0, 29 x 9 3 3 -  #/ (4−x)(2+x) ⎩ τ = −2   0,29 ≤ x ≤ 1 ABC. 9.  4             |GHZ $ |W B   x = 0     |W    4  ,    3 $ x = 1     |GHZ  4  ,    -7,   -//.

(34)     .     . .   .  .  .       .      .  . |GHZ  |W . F,   B  , %7              |GHZ $ |W  x = 0       |W  $ x = 1       |GHZ.

(35)          "  )         4   .  23    "            B |H     *  $ |V     "  .         "         M      01    B |H =. 1 , 0. |V =. 0 . 1. <       3  "        4 ,       % $     %      H  ,  "    ,    3               %3   7    $ )  )          $         .    .       .     "          % % 0 1         3      3 $         $   . .      . .      %  ,        4                 " 3   3     |ψ      $ U              "  |ψ   |ψ = U|ψ  ,. - /.  -UU−1 = I/B.     "  % . |ψ   = U−1 |ψ  .. . - /.

(36)        . . :                   -σx 3 σy $ σz /  . B 4 :     XB    . 4.   σx 0 1. X=. 1 0. -/. ,.                 B X|H = |V,. 4 :     YB &  . X|V = |H.. - /.   σy Y=. 0 −i i 0. ,. - /.                  4 π      e±i 2 = ±iB Y|H = i|V,. 4 :     ZB &  . Y|V = −i|H.. -/. 0 −1. -/.   σz Z=. 1 0. ,.      (           ,    |VB Z|H = |H,. Z|V = −|V.. -/. !           N        HB 1 H= √ 2. 1 1 1 −1. ,. -/.                   %     4 3           7B |H + |V √ , 2 |H + |V √ = |H, H 2. H|H =. |H − |V √ , 2 |H − |V √ H = |V. 2. H|V =. - #/ -. /.

(37)        .      . .    . !. F,   B 0 1 &         %. K                 "   "    I : 7,   -/            &         3 *$  $ B  J!K 4        J!K3    ⎛ ⎞ 1 0 0 0 ⎜ 0 1 0 0 ⎟ ⎜ ⎟ - / ⎝ 0 0 0 1 ⎠. 0 0 1 0         B           3   7,   -/3 $      ("   3      |H  ("   3        |V  ("     -  |H  " " |V  "  / :           2       4                  %3         .  .     .  .   !" .           0 1      7,                E      :   x -    /  y -   ("/      "  H $ V            $    E |βxy   .

(38)        . . F,   B 0 1 &       %       E. . (         |H    (" $ |V         √       H3   8   "    (|H − |V)/ 2 : ,          J!K3  ("     |V       |V   3        (" .        |H    (  √  |βV H  = (|HH − |VV)/ 2 :             B |HH + |VV √ , 2 |HV + |VH √ , |βHV  = 2 |HH − |VV √ , |βV H  = 2 |HV − |VH √ . |βV V  = 2. |βHH  =. E            .  . . - / -. /. -. /. - /.          .  !" .            -7,  /        |GHZ   "     x3 y $ z      "  H  V 3      |GHZxyz         7,  -/3  ( 3      x = H 3 y = H $ z =√H    7  x "     4  (|H + |V)/ 2 $ 8    7  y     √ J!K      (|HH + |VV)/ 2 : ,   x 8    z  √   ,  J!K     |GHZxyz  = (|HHH + |VVV)/ 2 ,    3            B 1 |GHZHHH  = √ (|HHH + |VVV), 2. - /.

(39)        . . . . F,  B 0 1 &        %      GHZ. 1 |GHZHHV  = √ (|HHV + |VVH), 2 1 |GHZHV H  = √ (|HVH + |VHV), 2 1 |GHZHV V  = √ (|HVV + |VHH), 2 1 |GHZV HH  = √ (|HHH − |VVV), 2 1 |GHZV HV  = √ (|HHV − |VVH), 2 1 |GHZV V H  = √ (HVH − |VHV), 2 1 |GHZV V V  = √ (|HVV − |VHH). 2. - / - / - #/ - / - / - / - /. <*         7,  -/                   y  "         2      J!K    x          x =√H 3 y = H $ z = H 3   x          (|H + |V)/ √2 $ 8    J!K     (|HH + |VV)/ 2 : ,   y    y 8  √z   J!K       |GHZxyz  = (|HHH + |VVV) 2 :         B 1 |GHZHHH  = √ (|HHH + |VVV), 2 1 |GHZHHV  = √ (|HHV + |VVH), 2 1 |GHZHV H  = √ (|HVV + |VHH), 2. -. /. - / - /.

(40)         1 |GHZHV V  = √ (|HVH + |VHV), 2 1 |GHZV HH  = √ (|HHH − |VVV), 2 1 |GHZV HV  = √ (|HHV − |VVH), 2 1 |GHZV V H  = √ (|HVV − |VHH), 2 1 |GHZV V V  = √ (|HVH − |VHV). 2.  - / - / -#/ - / - /. ,   (                            3  (    7,  -/         J!K        N  H   -/      J!K   8  z    J!K   8  y   2    N   .         I       "    )   .                .     . #   λ :  λ                        ,                ,     01    7              (     f - %/ $ s - /      " nf $ ns -7,   /   ( f    "( % %     ( s .            01            %,  Ψ         -( x  y /3 7,    :               %     exp −ins ωc l  0  , -/ 0 exp −ins ωc l  l   ,    3 ω       *  $ c  "       "  7      -     "  /  B ω Γ = l(ns − nf ), - / c $            B Φ=. 1ω l(ns + nf ). 2c. - /.

(41)        . #. F,   B 01 &     λ.          B e−iΦ. Γ. e−i 2 0. 0 Γ ei 2. -/. .. L       e−iΦ      01       4             *      .        ,        .               )               3 "          $ 7  )             :       B TΓ =. cos(−Ψ) − sin(−Ψ). sin(−Ψ) cos(−Ψ). Γ. e−i 2 0. 0 Γ ei 2. cos Ψ sin Ψ − sin Ψ cos Ψ. . -/.     λ      3  Γ                $  Ψ  %,                    % . (           λ/23  Γ = π 3        %   B Tπ = −i.      . cos(2Ψ) sin(2Ψ). sin(2Ψ) − cos(2Ψ). .. -/.      X   " Tπ. = −iX. -/.

(42)              % H X Z Y. Ψ π 8 π 4. 0     X $ Z    [Z; X] = 2iY.     B <,           λ/2          %  . cos(2Ψ) sin(2Ψ). sin(2Ψ) − cos(2Ψ) ⇒Ψ. = =. π . 8. 0 1 1 0. - #/ -. /. .          22, 5o       4  ,                    % -"   /. $.   .      (       $     t2 $  O(    r2 3      t2 + r2 = 1    . (    7        3     |03 $  O3     |13        B t|0 + r|1 √ . - / t2 + r 2. $.     .    0 #1         3  ( 3   |H  ## P    $  O(   |V  ## P   4       "    *   3    7       |0  ,   *    3         7    |1. %   &           "    2  "  -.5/3          -   ,  AI/  "    - , "/   5     *$  "   , $       ω     $ k       $ ω1 − k1 $ ω2 − k2     $   "      3  ω = ω1 + ω2 - /.

(43)         $. k = k1 + k2 .. -. /. N$    .5B  Q $  QQ :  Q         3    (    |HH + |VV √ . |ψ = - / 2 :  QQ           3  ( B |ψ =. |HV + |VH √ . 2. - /.   2            E *   4          λ <*       )  ( )         4   .   .  .     .  )           9       |GHZ 010 #1   (       % AI3        -K3 51 3 52 3 53 /3   λ/2    22,5o                  3    -E/3       -.!:E        *  $ O(  " / $     β 4 4  -EE!/ 4      .5  QQ  ( % **     -" 7,   /       ,     - $ /        EE!     ,    2       % AIB 1 |ψab = √ (|Ha |Vb − |Va |Hb ), 2. - /.   ,            3              .53    |ψ =. 1 (|Ha |Vb − |Va |Hb )(|H a |V b − |V a |H b ), 2. - /.      $        ,  $   4        .  "     GHZ              4                 4 B.

(44)        . . F,   B 0 1 '( )         |GHZ. 4   B F    *  -|Ha /B    "2  .!:E $ ,   K3     "    )        |Ha → |HT .. - /. F |Va B √    O(   .E $         (|V + |H)/ 2  2       λ/23  ,   .!:E *$ 50 %   4        O( $    51        $    52 3   |Va →. |V1 + |H2 √ . 2. - #/. 4   B F |Hb B  r2        O(   E $   51 3    2       $   53    4   t2 B |Hb → r2 |H1 +t2 |H3 . - / F |Vb B  2   . O(    r2 $      4.

(45)          t2   E3         52 3       53 B |Vb → r2 |V2 +t2 |V3 . - / .               3                )                  "      - %  /3                   ,      B rt − √ [|HT (|H 1 |H 2 |V3 + |V 1 |V2 |H 3 ) + 2 2 + |H T (|V1 |V 2 |H3 + |H1 |H2 |V 3 )].. - /. .      )       , B rt − √ |HT (|H1 |H2 |V3 + |V1 |V2 |H3 ), 2. - /.     |GHZ )     )    *   E 50/50 -r2 = 1/2, t2 = 1/2/ .         |GHZ   "       .      |GHZ       51 3 52 3 $ 53 -5T    ,, /    | + 45o  =. |H + |V √ 2. | − 45o  =. |H − |V √ . 2. -. /.   )     51        -          |GHZ    ,, /  " "B 1 |HT [| + 45o 1 (|H2 |V3 + |V2 |H3 ) + 2 + | − 45o 1 (|H2 |V3 − |V2 |H3 )].. . ( 3   . - /.     +45o  51     $. 1 √ |HT | + 45o 1 (|H2 |V3 + |V2 |H3 ). 2. - /. )        "  1 √ |HT | + 45o 1 (| + 45o 2 | + 45o 3 − | − 45o 2 | − 45o 3 ). 2. - /.  )        %       +45o  51 3   52    −45o $   53          +45o  −45o  :    , 7   7,  3    %   %.

(46)        . F,  B 01 &  )    (       |GHZ 5   52 *$    −45o 3  53   −45o -  /    +45o -  /   7,  /    51 *$    +45o $  /     0o .            a3 $  , 7   "  4          53    7,  -/       "      |GHZ           −45o  N*   7            +45o  51 $ −45o  52 $           +45o  53  K 2            a3   3     "(     a $ b  ,  $   $    4   % .  7  3  7,  -/            51    0o -(   /H    |GHZ  $         (   |V2 |H3      3 * ,     +45o $ −45o  53 .

(47)  .  .    (                B GHZ $ W         GHZ                           $   3       4         4    3   W               %                  $3           $              *    W , 3                         3    (    .&3 (           $          .    3           $        .&3 %           .&            )    .& .   )        *$        3                             )     (  - GHZ 3 W $   )/      .      $    )         " *     %                     %                   N*   "      "           8         %      3            E    J!K $       GHZ         J!K. .

(48)                           %                   %     0 1 $  0 1       |GHZ      (3 <3 E $ R3   $          1 |GHZ = √ (|H2 |H3 |H4 + |V2 |V3 |V4 ), -< / 2   8     8       <    23 E  3 $ R  4   |ψ         <B |ψ = α|H1 + β|V1 .. -< /.              7,  <              B |ψ0. = |ψ|GHZ 1 = √ [α|H1 (|H2 |H3 |H4 + |V2 |V3 |V4 ) + 2 +β|V1 (|H2 |H3 |H4 + |V2 |V3 |V4 )].. -</. <* 3 <          "2   J!K3                   GHZ $    R   "% B 1 |ψ1 = √ [α|H1 (|H2 |H3 |H4 + |V2 |V3 |V4 ) + 2 -< / + β|V1 (|V2 |V3 |H4 + |H2 |H3 |V4 )]. : ,.          "2   N  B 1 [α(|H1 + |V1 )(|H2 |H3 |H4 + |V2 |V3 |V4 ) + |ψ2 = 2 + β(|H1 − |V1 )(|V2 |V3 |H4 + |H2 |H3 |V4 )]. . -< /.

(49)  .          . . F,  < B 0 1      '  < |HH |HV |VH |VV. '  R |H |V |H |V.   E α|H4 + β|V4 α|V4 + β|H4 α|H4 − β|V4 α|V4 − β|H4.    < B      E    R.   < $.        ,                      GHZ B |ψ2. =. 1 √ [|H1 |H2 |H3 (α|H4 + β|V4 ) + 2 + |H1 |V2 |V3 (α|V4 + β|H4 ) + + |V1 |H2 |H3 (α|H4 − β|V4 ) + + |V1 |V2 |V3 (α|V4 − β|H4 )].. -</. .      < $ R          $ "     E     "   ,                   |φ :  <    E ,8    < $ R      |HHH  E (    3     |HVV E      X3     |VHH E       "2       Z3      |VVV E              X $  ,  Z       5                   E       $    $   :            7      !   %  .

(50)  .          . .  "  % - ,   %/3    $       3                2        "    (.

(51) . . < 5 <   3      %3 :  & ,  &$3   $           <    3   M,* & : * 3          "        $  ,  < 5    F   A"   : <  E,% - / <  A"    *$  L,S -/ $   2  ,4   ?K* "* Q     T   3 '      ,@3     ,      " $        ( <      5     J "  E,% - /3   $   $            . #.

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