Nathaniel: Created page with "{{Function |name=InSeparableBall |desc=Checks whether or not an operator is in the ball of separability centered at the maximally-mixed state |rel=IsAbsPPT
IsSepara..."

2014-11-15T00:55:43Z

Created page with "{{Function |name=InSeparableBall |desc=Checks whether or not an operator is in the ball of separability centered at the maximally-mixed state |rel=IsAbsPPT<br />IsSepara..."

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{{Function
|name=InSeparableBall
|desc=Checks whether or not an operator is in the ball of separability centered at the maximally-mixed state
|rel=[[IsAbsPPT]]<br />[[IsSeparable]]
|cat=[[List of functions#Ball_of_separability|Ball of separability]]
|upd=November 14, 2014
|v=0.60}}
<tt>'''InSeparableBall'''</tt> is a [[List of functions|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 <ref>L. Gurvits and H. Barnum. Largest separable balls around the maximally mixed bipartite quantum state. ''Phys. Rev. A'', 66:062311, 2002. E-print: [http://arxiv.org/abs/quant-ph/0204159 arXiv:quant-ph/0204159]</ref>.

==Syntax==
* <tt>ISB = InSeparableBall(X)</tt>

==Argument descriptions==
* <tt>X</tt>: 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 $m = n = 2$:
<syntaxhighlight>
>> 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
</syntaxhighlight>

{{SourceCode|name=InSeparableBall}}

==References==
<references />

InSeparableBall - Revision history

Nathaniel: Created page with "{{Function |name=InSeparableBall |desc=Checks whether or not an operator is in the ball of separability centered at the maximally-mixed state |rel=IsAbsPPTIsSepara..."

Nathaniel: Created page with "{{Function |name=InSeparableBall |desc=Checks whether or not an operator is in the ball of separability centered at the maximally-mixed state |rel=IsAbsPPT
IsSepara..."