Integral representations of separable states *

We study a separability problem suggested by mathematical description of bipartite quantum systems. We consider hermitian 2-forms on the tensor product H = K ⊗ L, where K, L are finite-dimensional complex spaces. Such a form is called separable if it is a convex combination of hermitian tensor products σ*p⊙σp of 1-forms σp on H that are product forms σp = ϕp ⊗ ψp, where ϕp ∈ K*, ψpL*.We introduce an integral representation of separable forms. We show that the integral of Dz*Φ* ⊙ Dz*Φ of any square integrable map Φ : ℂn → ℂm, with square integrable conjugate derivative Dz*Φ, is a separable form. Conversely, any separable form in the interior of the set of such forms can be represented in this way. This implies that any separable mixed state (and only such states) can be either explicitly represented in the integral form, or it may be arbitrarily well approximated by such states.