Semidefinite descriptions of low-dimensional 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 = ∑ l x l ⊗ x l ′ , where x l ∈ K and x l ′ ∈ 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 ∈ 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 m ⩽ 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.