Separability rigidity of Gaussian states under positive domination

Abstract

We prove a separability rigidity theorem for finite-mode Gaussian states. Let ${\rho }_G$ be Gaussian and let $\mathcal{I}$ be a fixed partition of the modes. If ${\rho }_G$ dominates a nonzero positive operator $T$ that is separable across $\mathcal{I}$, $0\ne T\preccurlyeq {\rho }_G$, then ${\rho }_G$ itself is separable across $\mathcal{I}$. The dominated operator is arbitrary and need not be Gaussian. As consequences, convex mixing among finitely many separability partitions creates no new Gaussian states: a Gaussian state belonging to such a convex class is already separable across one fixed partition. In particular, full inseparability and genuine multipartite entanglement coincide for finite-mode Gaussian states, and identical copies of a biseparable Gaussian state cannot activate genuine multipartite entanglement. We also show that the implication cannot be reversed by constructing a fully separable Gaussian state that dominates a positive multiple of a genuine multipartite entangled state.

22 pages