ChessboardState

ChessboardState
Produces a chessboard state
Other toolboxes required	IsPPT IsPSD opt_args PartialTranspose PermuteSystems

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