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

TL;DR

This paper introduces a new symplectic-invariant measure of correlations in Gaussian systems, based on partner-mode overlaps, providing a geometric interpretation and a criterion for entanglement detection.

Contribution

It presents a novel correlation quantifier for Gaussian states that is symplectic-invariant and geometrically interpretable, extending previous overlap measures and offering an entanglement criterion.

Findings

Defines a new correlation measure $\\mathcal{D}^{\mathrm{sym}}$ for Gaussian states.

Provides a necessary and sufficient criterion for two-mode entanglement.

Demonstrates the measure with a scalar field in Minkowski spacetime.

Abstract

We introduce a locally symplectic-invariant quantifier of correlations between two different arbitrary modes in bosonic Gaussian systems, denoted by $\mathcal{D}^{\mathrm{sym}}$. 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 $\mathcal{D}^{\mathrm{sym}}$, 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.