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Semidefinite descriptions of low-dimensional separable matrix cones. (English) Zbl 1143.15022

Summary: Let \(K\subset E\), \(K'\subset E'\) be convex cones residing in finite-dimensional real vector spaces. An element \(y\) in the tensor product \(E\otimes E'\) is \(K\otimes K'\)-separable if it can be represented as finite sum \(y= \sum_l x_l\otimes x_l'\) where \(x_l\in K\) and \(x_l'\in K'\) for all \(l\). Let \({\mathcal S}(n)\), \({\mathcal H}(n)\), \({\mathcal Q}(n)\) be the spaces of \(n\times n\) real symmetric, complex Hermitian and quaternionic Hermitian matrices, respectively. Let further \(S_+(n)\), \(H_+(n)\), \(Q_+(n)\) be the cones of positive semidefinite matrices in these spaces.
If a matrix \(A\in{\mathcal H}(mn)={\mathcal H}(m)\otimes{\mathcal H}(n)\) is \(H_+(m)\otimes H_+(n)\)-separable, then it fulfills also the so-called PPT condition, i.e. it is positive semidefinite and has a positive semidefinite partial transpose. The same implication holds for matrices in the spaces \({\mathcal S}(m)\otimes{\mathcal S}(n)\), \({\mathcal H}(m)\otimes{\mathcal S}(n)\), and for \(m\leq 2\) in the space \({\mathcal Q}(m)\otimes{\mathcal S}(n)\).
We provide a complete enumeration of all pairs \((n,m)\) when the inverse implication is also true for each of the above spaces, i.e. the PPT condition is sufficient for separability. We also show that a matrix in \({\mathcal S}(n)\otimes{\mathcal S}(2)\) is \(Q_+(n)\otimes S_+(2)\)-separable if and only if it is positive semidefinite.

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
15B57 Hermitian, skew-Hermitian, and related matrices
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