Full Inseparability and Genuine Multipartite Entanglement Coincide for Finite-Mode Gaussian States

Abstract

For general mixed states, entanglement across every bipartition need not imply genuine multipartite entanglement (GME), because a biseparable decomposition may switch the separable cut from term to term. We prove that this convex ambiguity disappears for Gaussian states of finitely many bosonic modes. More generally, for any finite family of partitions, a Gaussian density operator in the trace-norm-closed convex class generated by states separable across those partitions is already separable across one fixed partition in the family. Only the target is Gaussian; a valid decomposition may be continuous and may contain arbitrary non-Gaussian states. Thus full inseparability and GME coincide, Gaussian k-separability and k-producibility reduce to fixed-partition tests, and party-wise tensor powers cannot activate GME from a biseparable Gaussian state. The proof combines a spectral selector with a holomorphic rigidity argument that converts one product vector in the square-root range of a Gaussian state into a block-local covariance certificate. The result shows that partition mixing, a generic mixed-state mechanism, adds no new exact finite-mode Gaussian states.