Hierarchies of Gaussian multimode entanglement from thermodynamic quantifiers

TL;DR

This paper introduces a thermodynamic approach to quantify and hierarchically characterize multimode entanglement in Gaussian quantum systems using work extraction metrics, establishing new entanglement measures and bounds.

Contribution

It develops a thermodynamic framework linking work extraction to Gaussian multimode entanglement, introducing the $k$-ergotropic score as a novel quantifier.

Findings

The 2-local ergotropic gap is a faithful entanglement monotone.

The $k$-ergotropic score quantifies multimode entanglement across partitions.

Closed-form relations are derived for three-mode Gaussian states.

Abstract

We develop a thermodynamic characterization of multimode entanglement in pure continuous-variable systems by quantifying the gap between globally and locally extractable work (ergotropy). For arbitrary pure multimode Gaussian states, we prove that the $2$-local ergotropic gap is a faithful entanglement monotone across any bipartition and constitutes a functionally independent upper bound to the Renyi-2 entanglement entropy. We further introduce the $k$-ergotropic score, the minimum $k$-local ergotropic gap, and show that it faithfully quantifies multimode entanglement across $k$ partitions. For pure three-mode Gaussian states, we derive its closed-form relation with the geometric measure for genuine multimode entanglement $(k=2)$, and total Gaussian multimode entanglement $(k=3)$. For systems with more than three modes, the $k$-ergotropic score becomes a functionally independent measure of multimode entanglement to the standard geometric measures. Our results reveal a direct operational hierarchy linking Gaussian multimode entanglement to work extraction under locality constraints, and provide a computable and experimentally accessible thermodynamic framework for characterizing quantum correlations.