Partner-mode overlap as a symplectic-invariant measure of correlations in Gaussian Systems

We introduce a locally symplectic-invariant quantifier of correlations between two different arbitrary modes in bosonic Gaussian systems, denoted by ${\mathcal{D}}^{\mathrm{s}\mathrm{y}\mathrm{m}}$. This quantity admits a simple geometric interpretation as an overlap between each mode and the purification partner of the other, formulated using the complex-structure description of Gaussian states. The construction builds on the partner-mode framework of Ref.~\cite{agullo_correlation_2025} and can be viewed as a symmetrized extension of earlier overlap-based measures~\cite{osawa2025entanglement}. We formulate a simple necessary and sufficient criterion for two-mode entanglement in Gaussian states in terms of ${\mathcal{D}}^{\mathrm{s}\mathrm{y}\mathrm{m}}$, placing on firm quantitative footing the intuition that entanglement with a given localized mode `lives’ on the spatial support of its partner mode. We illustrate the framework with a numerical analysis of a scalar field in Minkowski spacetime and discuss its extension to multimode systems and mixed Gaussian states.