Integral representations of separable states

Bronisław Jakubczyk, Gabriel Pietrzkowski

Published 2008-02-03, updated 2008-12-07Version 2

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.