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Produces a chessboard state

Other toolboxes required none
Related functions BreuerState
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.


  • 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.


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 =


>> IsPPT(rho)

ans =


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 here to view this function's source code on github.


  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