HorodeckiState

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HorodeckiState
Produces a Horodecki state

Other toolboxes required none
Related functions BreuerState
ChessboardState
Function category Special states, vectors, and operators

HorodeckiState is a function that produces a "Horodecki" bound entangled state in either $M_3 \otimes M_3$ (two-qutrit space) or $M_2 \otimes M_4$. These states were defined in [1] and have the following standard basis representation: \[\rho_a^{3\otimes 3} := \frac{1}{8a+1}\begin{bmatrix}a & 0 & 0 & 0 & a & 0 & 0 & 0 & a \\ 0 & a & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 \\ a & 0 & 0 & 0 & a & 0 & 0 & 0 & a \\ 0 & 0 & 0 & 0 & 0 & a & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \tfrac{1}{2}(1+a) & 0 & \tfrac{1}{2}\sqrt{1-a^2} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & a & 0 \\ a & 0 & 0 & 0 & a & 0 & \tfrac{1}{2}\sqrt{1-a^2} & 0 & \tfrac{1}{2}(1+a)\end{bmatrix}\] and \[\rho_a^{2 \otimes 4} := \frac{1}{7a+1}\begin{bmatrix}a & 0 & 0 & 0 & 0 & a & 0 & 0 \\ 0 & a & 0 & 0 & 0 & 0 & a & 0 \\ 0 & 0 & a & 0 & 0 & 0 & 0 & a \\ 0 & 0 & 0 & a & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \tfrac{1}{2}(1+a) & 0 & 0 & \tfrac{1}{2}\sqrt{1-a^2} \\ a & 0 & 0 & 0 & 0 & a & 0 & 0 \\ 0 & a & 0 & 0 & 0 & 0 & a & 0 \\ 0 & 0 & a & 0 & \tfrac{1}{2}\sqrt{1-a^2} & 0 & 0 & \tfrac{1}{2}(1+a)\end{bmatrix}.\]

Syntax

  • HORO_STATE = HorodeckiState(A)
  • HORO_STATE = HorodeckiState(A,DIM)

Argument descriptions

  • A: A real number between 0 and 1 that determines which Horodecki state is produced.
  • DIM (optional, default [3,3]): The dimensions of the subsystems that the state should act on. Must be one of [3,3] or [2,4].

Examples

Two-qutrit bound entangled state

The following code generates a two-qutrit Horodecki state and verifies that it is bound entangled by checking that it has positive partial transpose and is not separable:

>> rho = HorodeckiState(0.5)
 
rho =
 
    0.1000         0         0         0    0.1000         0         0         0    0.1000
         0    0.1000         0         0         0         0         0         0         0
         0         0    0.1000         0         0         0         0         0         0
         0         0         0    0.1000         0         0         0         0         0
    0.1000         0         0         0    0.1000         0         0         0    0.1000
         0         0         0         0         0    0.1000         0         0         0
         0         0         0         0         0         0    0.1500         0    0.0866
         0         0         0         0         0         0         0    0.1000         0
    0.1000         0         0         0    0.1000         0    0.0866         0    0.1500
 
>> IsPPT(rho)
 
ans =
 
     1
 
>> IsSeparable(rho)
Determined to be entangled via the realignment criterion. Reference:
K. Chen and L.-A. Wu. A matrix realignment method for recognizing entanglement. Quantum Inf. Comput., 3:193-202, 2003.
 
ans =
 
     0

A (2 ⊗ 4)-dimensional bound entangled state

The following code generates a Horodecki state in $M_2 \otimes M_4$ and verifies that it is bound entangled by checking that it has positive partial transpose and is not separable:

>> rho = HorodeckiState(0.5,[2,4])
 
rho =
 
    0.1111         0         0         0         0    0.1111         0         0
         0    0.1111         0         0         0         0    0.1111         0
         0         0    0.1111         0         0         0         0    0.1111
         0         0         0    0.1111         0         0         0         0
         0         0         0         0    0.1667         0         0    0.0962
    0.1111         0         0         0         0    0.1111         0         0
         0    0.1111         0         0         0         0    0.1111         0
         0         0    0.1111         0    0.0962         0         0    0.1667
 
>> IsPPT(rho,2,[2,4])
 
ans =
 
     1
 
>> IsSeparable(rho,[2,4])
Determined to be entangled by not having a 2-copy PPT symmetric extension. Reference:
A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri. A complete family of separability criteria. Phys. Rev. A, 69:022308, 2004.
 
ans =
 
     0

Source code

Click on "expand" to the right to view the MATLAB source code for this function.

  1. %%  HORODECKISTATE    Produces a Horodecki state
  2. %   This function has one required input argument:
  3. %     A: a real parameter in [0,1]
  4. %
  5. %   HORO_STATE = HorodeckiState(A) returns the 3x3 bound entangled
  6. %   Horodecki state described in [1].
  7. %
  8. %   This function has one optional input argument:
  9. %     DIM (default is [3,3], but can be either [3,3] or [2,4])
  10. %
  11. %   HORO_STATE = HorodeckiState(A,DIM) returns the Horodecki state in
  12. %   either (3 \otimes 3)-dimensional space, or (2 \otimes 4)-dimensional
  13. %   space, depending on the dimensions in the 1-by-2 vector DIM.
  14. %
  15. %   The Horodecki state was introduced in [1] which serves as an example in 
  16. %   C^3 \otimes C^3 or C^2 \otimes C^4 of an entangled state that is
  17. %   positive under partial transpose (PPT). The state is PPT for all 
  18. %   a \in [0,1], and separable only for a = 0 or a = 1.
  19. %
  20. %   Note: Refer to [2] (specifically equations (1) and (2)) for more 
  21. %   information on this state and its properties. The 3x3 Horodecki state 
  22. %   is defined explicitly in Section 4.1 of [1] and the 2x4 Horodecki state 
  23. %   is defined explicitly in Section 4.2 of [1].
  24. %
  25. %   References:
  26. %   [1] P. Horodecki. Separability criterion and inseparable mixed states
  27. %       with positive partial transposition. E-print: 
  28. %       arXiv:quant-ph/9703004, 1997.
  29. %       
  30. %   [2] K. Chruscinski. On the symmetry of the seminal Horodecki state.
  31. %       E-print: arXiv:1009.4385 [quant-ph], 2010.
  32. %
  33. %   URL: http://www.qetlab.com/HorodeckiState
  34.  
  35. %   requires: opt_args.m
  36. %   authors: Vincent Russo (vrusso@uwaterloo.ca)
  37. %            Nathaniel Johnston (nathaniel@njohnston.ca)
  38. %   package: QETLAB 
  39. %   last updated: December 15, 2014
  40.  
  41. function horo_state = HorodeckiState( a, varargin )
  42.  
  43. % set optional argument defaults: dim = [3,3]
  44. [dim] = opt_args({ [3,3] },varargin{:});
  45.  
  46. if a < 0 || a > 1
  47.     error('HorodeckiState:InvalidA','Argument A must be in the interval [0,1].');
  48. end
  49.  
  50. if isequal(dim(:),[3;3])
  51.     N_a = 1/(8*a+1);
  52.     b = (1+a)/2;
  53.     c = sqrt(1-a^2)/2;
  54.  
  55.     horo_state = N_a * [ a 0 0 0 a 0 0 0 a;
  56.                          0 a 0 0 0 0 0 0 0;
  57.                          0 0 a 0 0 0 0 0 0;
  58.                          0 0 0 a 0 0 0 0 0;
  59.                          a 0 0 0 a 0 0 0 a;
  60.                          0 0 0 0 0 a 0 0 0;
  61.                          0 0 0 0 0 0 b 0 c;
  62.                          0 0 0 0 0 0 0 a 0;
  63.                          a 0 0 0 a 0 c 0 b ];
  64.  
  65. elseif isequal(dim(:),[2;4])
  66.     N_a = 1/(7*a+1);
  67.     b = (1+a)/2;
  68.     c = sqrt(1-a^2)/2;
  69.  
  70.     horo_state = N_a * [ a 0 0 0 0 a 0 0;
  71.                          0 a 0 0 0 0 a 0;
  72.                          0 0 a 0 0 0 0 a;
  73.                          0 0 0 a 0 0 0 0;
  74.                          0 0 0 0 b 0 0 c;
  75.                          a 0 0 0 0 a 0 0;
  76.                          0 a 0 0 0 0 a 0;
  77.                          0 0 a 0 c 0 0 b ];
  78. else
  79.     error('HorodeckiState:InvalidDim','DIM must be one of [3,3] or [2,4].');
  80. end

References

  1. P. Horodecki. Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A, 232:333, 1997. E-print: arXiv:quant-ph/9703004