# ChessboardState

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### When specifying

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

## References

- ↑
^{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