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

TL;DR

This work introduces $\mathcal{D}^{\mathrm{sym}}$, a locally symplectic-invariant overlap between two Gaussian modes, defined via their purification partners in the complex-structure formalism. It proves a clear necessary-and-sufficient entanglement criterion for two local Gaussian modes, $\mathcal{D}^{\mathrm{sym}} > \mathcal{D}_c$, expressible solely through symplectic invariants and valid for mixed states as well. The framework is extended to quantum field theory and demonstrated with a ball–shell example in Minkowski space, where $\mathcal{D}^{\mathrm{sym}}$ tracks entanglement distribution and relates to the logarithmic negativity in the weak-entanglement regime. A key practical insight is that, for weakly entangled localized modes, $E_{\mathcal N}$ is well-approximated by a linear function of the symmetric overlap deficit, facilitating quantitative entanglement assessments in field-theoretic settings. The results provide a coordinate-free, geometrically intuitive lens on how entanglement is distributed across space and modes in Gaussian quantum systems.

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.

Figures (11)

• Figure 1: Entanglement between $A$ and $B$ as a function of the Beam splitter parameter $\theta$ for a squeezer parameter $r=0.5$. The blue and orange lines describes the Logarithmic Negativity $E_{\mathcal{N}}$ and the difference $\mathcal{D}^{sym} - \mathcal{D}_{c}$, respectively. The purple vertical lines represent the values for $\theta$ where $A$ and $B$ are not correlated and the state is separable.
• Figure 2: Illustration of the regions of support for the field modes considered in this example. Mode $A$ is supported within a ball of radius $R_A$ (light orange). Mode $B$ is supported within a spherical shell surrounding the ball, with inner radius $R_B$ and radial width $d_B$ (dark blue).
• Figure 3: Logarithmic negativity between modes $A$ and $B$, supported in a ball and a concentric spherical shell respectively, as a function of their radial separation $R_B-R_A$. The parameters are $m=0$ and $d_B=0.5$. All lengths are measured in units of $R_A$.
• Figure 4: Logarithmic negativity as a function of the dimensionless field mass $\mu = m R_A$. The parameters are $R_B=R_A$ (no gap between the supports of the modes) and $d_B=0.5R_A$.
• Figure 5: Logarithmic negativity as a function of the shell width $d_B$ (measured in units of $R_A$) for different values of the dimensionless field mass $\mu=m R_A$. The parameters are $R_B=R_A$. Entanglement is non-zero only for a finite range of values of $d_B$ubiquitous.