ChessboardState

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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 [1]. 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 [1], 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 [1]. 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.
  24.  
  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
  31.  
  32. function rho = ChessboardState(a,b,c,d,m,n,varargin)
  33.  
  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{:});
  36.  
  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];
  41.  
  42. rho = v1'*v1 + v2'*v2 + v3'*v3 + v4'*v4;
  43. rho = rho/trace(rho);
  44.  
  45. if(~IsPPT(rho))
  46.     warning('ChessboardState:NotPPT','The specified chessboard state does not have positive partial transpose.');
  47. end

References

  1. 1.0 1.1 1.2 D. Bruss and A. Peres. Construction of quantum states with bound entanglement. Phys. Rev. A, 61:30301(R), 2000. E-print: arXiv:quant-ph/9911056