Semidefinite descriptions of separable matrix cones

Abstract

Let K E, K′ E′ be convex cones residing in finite-dimensional real vector spaces. An element y in the tensor product E E′ is K K′-separable if it can be represented as finite sum y = Pl xl x′ l, where xl 2 K and x′ l 2 K′ for all l. Let S(n), H(n), Q(n) be the spaces of n × 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 2 H(mn) = H(m) H(n) is H+(m) 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 S(m) S(n), H(m) S(n), and for < 2 in the space Q(m) 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 Q(n) S(2) is Q+(n) S+(2)-separable if and only if it is positive semidefinite.