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arXiv:2512.18410

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

Ivan Agullo, Eduardo Martín-Martínez, Sergi Nadal-Gisbert, Koji Yamaguchi

quant-phgr-qchep-thSubmitted Dec 20, 2025• Announced Dec 23, 2025• 0 citations
v1 · Dec 20, 2025v2 · Jul 23, 2026
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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.

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22 pages, 11 figures

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We present a derivation of the Von Neumann entropy and mutual information of arbitrary two-mode Gaussian states, based on the explicit determination of the symplectic eigenvalues of a generic covariance matrix. The key role of the symplectic invariants in such a determination is pointed out. We show that the Von Neumann entropy depends on two symplectic invariants, while the purity (or the linear entropy) is determined by only one invariant, so that the two quantities provide two different hierarchies of mixed Gaussian states. A comparison between mutual information and entanglement of formation for symmetric states is considered, taking note of the crucial role of the symplectic eigenvalues in qualifying and quantifying the correlations present in a generic state.
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