Hierarchies of Gaussian multimode entanglement from thermodynamic quantifiers

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.