# InSeparableBall

 Other toolboxes required InSeparableBall Checks whether or not an operator is in the ball of separability centered at the maximally-mixed state none IsAbsPPTIsSeparable Ball of separability

InSeparableBall is a function that determines whether or not a density matrix is contained within the ball of states that are separable centered at the maximally-mixed state (more generally, it determines whether or not a positive semidefinite operator is within the ball of separability centered at an appropriately-scaled identity matrix). The size of this ball of separability was computed in [1].

## Syntax

• ISB = InSeparableBall(X)

## Argument descriptions

• X: A bipartite density matrix (or any bipartite positive semidefinite operator).

## Examples

The only states acting on $\mathbb{C}^m \otimes \mathbb{C}^n$ in the separable ball that do not have full rank are those with exactly 1 zero eigenvalue, and the $mn-1$ non-zero eigenvalues equal to each other. The following code highlights this fact when $m = n = 2$:

>> U = RandomUnitary(4);
>> lam = [1,1,1,0]/3;
>> rho = U*diag(lam)*U'; % random density matrix with eigenvalues [1,1,1,0]/3
>> InSeparableBall(rho)

ans =

1

>> lam2 = [1.01,1,0.99,0]/3;
>> rho2 = U*diag(lam2)*U'; % random density matrix with eigenvalues [1.01,1,0.99,0]/3
>> InSeparableBall(rho2)

ans =

0

## References

1. L. Gurvits and H. Barnum. Largest separable balls around the maximally mixed bipartite quantum state. Phys. Rev. A, 66:062311, 2002. E-print: arXiv:quant-ph/0204159