Partner-mode overlap as a symplectic-invariant measure of correlations in Gaussian quantum field theories

Abstract

We introduce a locally symplectic-invariant quantifier of correlations between arbitrary bosonic Gaussian modes, with particular emphasis on quantum field theory. The quantity, denoted by~${\mathcal{D}}^{\mathrm{sym}}$, admits a simple geometric interpretation as the symmetric overlap between each mode and the purification partner of the other, providing a direct geometric characterization of how correlations are distributed between modes. We derive a necessary and sufficient criterion for two-mode Gaussian entanglement in terms of ${\mathcal{D}}^{\mathrm{sym}}$, placing on firm quantitative footing the intuition that the entanglement of a localized mode is encoded in the spatial support of its purification partner. We demonstrate the framework for wavepacket modes of a scalar quantum field in Minkowski spacetime (illustrating how the geometry of partner modes reveals the spatial structure of quantum correlations) and discuss extensions to multimode systems and mixed Gaussian states.

22 pages, 11 figures