Integral representations of separable states

TL;DR

This paper introduces an integral representation for separable states in bipartite quantum systems, showing that all such states can be explicitly or approximately expressed through these integral forms, advancing understanding of quantum state separability.

Contribution

It provides a novel integral representation for separable forms, linking them to square integrable maps and their derivatives, and characterizes separable states via this integral framework.

Findings

Integral representation of separable forms established.

Separable states can be explicitly or approximately represented by the integral form.

Characterization of the interior of the set of separable forms achieved.

Abstract

We study a separability problem suggested by mathematical description of bipartite quantum systems. We consider Hermitian 2-forms on the tensor product $H=K\otimes L$, where $K,L$ are finite dimensional complex spaces. Inspired by quantum mechanical terminology we call such a form separable if it is a convex combination of hermitian tensor products $(\sigma_p)^*\odot \sigma_p$ of 1-forms $\sigma_p$ on $H$ that are product forms $\sigma_p=\phi_p\otimes \psi_p$, where $\phi_p\in K^*$, $\psi_p\in L^*$. We introduce an integral representation of separable forms. In particular, we show that the integral of $(D_{z^*}}\Phi)^*\odot D_{z^*}\Phi$ of any square integrable map $\Phi:\C^n\to \C^m$, with square integrable conjugate derivative $D_{z^*}\Phi$, is a separable form. Vice versa, 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.