ChessboardState

ChessboardState
Produces a chessboard state
Other toolboxes required	none
Related functions	BreuerState IsPPT PartialTranspose
Function category	Special states, vectors, and operators

ChessboardState is a function that produces a two-qutrit "chessboard state", as defined in . These states are of interest because they are bound entangled.

Syntax

• RHO = ChessboardState(A,B,C,D,M,N)
• RHO = ChessboardState(A,B,C,D,M,N,S,T)

Argument descriptions

• A,B,C,D,M,N: Six parameters that define chessboard states, as in , with S = A*conj(C)/conj(N) and T = A*D/M. If C*M*conj(N) does not equal A*B*conj(C) then RHO is entangled. If each of A,B,C,D,M,N are real then RHO has positive partial transpose, and is hence bound entangled.
• S,T: Additional (optional) parameters of the chessboard states, also as in . Note that, for certain choices of S and T, this state will not have positive partial transpose, and thus may not be bound entangled – a warning will be produced in these cases.

Examples

Generating bound entangled states

Chessboard states are useful because they form a wide family of bound entangled states. The following code generates a random chessboard state and verifies that it is entangled yet positive-partial-transpose (and hence bound entangled).

>> rho = ChessboardState(randn(1),randn(1),randn(1),randn(1),randn(1),randn(1));
>> IsSeparable(rho)
Determined to be entangled via the Filter Covariance Matrix Criterion. Reference:
O. Gittsovich, O. Gühne, P. Hyllus, and J. Eisert. Unifying several separability
conditions using the covariance matrix criterion. Phys. Rev. A, 78:052319, 2008.
 
ans =
 
     0
 
>> IsPPT(rho)
 
ans =
 
     1

When specifying S and T

If you specify S and T manually, it is possible that the resulting state will not have positive partial transpose – a warning is produced in these cases.

>> rho = ChessboardState(1,2,3,4,5,6,7,8);
Warning: The specified chessboard state does not have positive partial transpose. 
> In ChessboardState at 45

Source code

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

1. %%  CHESSBOARDSTATE    Produces a chessboard state

2. %   This function has six required arguments:

3. %     A,B,C,D,M,N: parameters of the chessboard state, as in [1]

4. %

5. %   RHO = ChessboardState(A,B,C,D,M,N) is the chessboard state defined in

6. %   [1], with S = A*conj(C)/conj(N) and T = A*D/M. If C*M*conj(N) ~=

7. %   A*B*conj(C) then RHO is entangled. If each of A,B,C,D,M,N are real then

8. %   RHO has positive partial transpose, and is hence bound entangled.

9. %

10. %   This function has two optional arguments:

11. %     S (default A*conj(C)/conj(N))

12. %     T (default A*D/M)

13. %   

14. %   RHO = ChessboardState(A,B,C,D,M,N,S,T) is the chessboard state defined

15. %   in [1]. Note that, for certain choices of S and T, this state will not

16. %   have positive partial transpose, and thus may not be bound entangled --

17. %   a warning will be produced in these cases.

18. %

19. %   URL: http://www.qetlab.com/ChessboardState

20. %

21. %   References:

22. %   [1] D. Bruss and A. Peres. Construction of quantum states with bound

23. %       entanglement. Phys. Rev. A, 61:30301(R), 2000.

25. %   requires: IsPPT.m, IsPSD.m, opt_args.m, PartialTranspose.m,

26. %             PermuteSystems.m

27. %

28. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)

29. %   package: QETLAB

30. %   last updated: March 13, 2013

32. function rho = ChessboardState(a,b,c,d,m,n,varargin)

34. % set optional argument defaults: s = ac*/n*, t = ad/m

35. [s,t] = opt_args({ a*conj(c)/conj(n), a*d/m },varargin{:});

37. v1 = [m,0,s,0,n,0,0,0,0];

38. v2 = [0,a,0,b,0,c,0,0,0];

39. v3 = [conj(n),0,0,0,-conj(m),0,t,0,0];

40. v4 = [0,conj(b),0,-conj(a),0,0,0,d,0];

42. rho = v1'*v1 + v2'*v2 + v3'*v3 + v4'*v4;

43. rho = rho/trace(rho);

45. if(~IsPPT(rho))

46.     warning('ChessboardState:NotPPT','The specified chessboard state does not have positive partial transpose.');

47. end