Separability rigidity of Gaussian states under positive domination

Abstract

We prove a separability rigidity theorem for finite-mode Gaussian states. Let $ρ_G$ be Gaussian and let $\mathcal I$ be a fixed partition of the modes. If $ρ_G$ dominates a nonzero positive operator $T$ that is separable across $\mathcal I$, $0\neq T\preccurlyeq ρ_G$, then $ρ_G$ 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.