# HorodeckiState

 Other toolboxes required HorodeckiState Produces a Horodecki state none BreuerStateChessboardState 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