Semidefinite descriptions of low-dimensional separable matrix cones Original Research Article

Let Ksubset ofE, K′subset ofE′ be convex cones residing in finite-dimensional real vector spaces. An element y in the tensor product Ecircle times operatorE′ is Kcircle times operatorK′-separable if it can be represented as finite sum image, where xlset membership, variantK and image for all l. Let image, image, image 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 image is H+(m)circle times operatorH+(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 image, image, and for mless-than-or-equals, slant2 in the space image. 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 image is Q+(n)circle times operatorS+(2)- separable if and only if it is positive semidefinite.