Ergotropic characterization of continuous variable entanglement

Continuous-variable quantum thermodynamics in the Gaussian regime provides a promising framework for investigating the energetic role of quantum correlations, particularly in optical systems. In this work, we introduce an entropy-free criterion for entanglement detection in bipartite Gaussian states, rooted in a distinct thermodynamic quantity: ergotropy–the maximum extractable work via unitary operations. By defining the relative ergotropic gap, which quantifies the disparity between global and local ergotropy, we derive two independent analytical bounds that distinguish entangled from separable states. These bounds coincide for a broad class of quantum states, making the criterion both necessary and sufficient in such cases. Unlike entropy-based measures, our ergotropic approach captures fundamentally different aspects of quantum correlations and entanglement, particularly in mixed continuous-variable systems. We also extend our analysis beyond the Gaussian regime to certain non-Gaussian states and observe that Gaussian ergotropy continues to reflect thermodynamic signatures in entangled states, albeit with some limitations. These findings establish a direct operational link between entanglement and energy storage, offering an experimentally accessible approach to entanglement detection in continuous-variable optical platforms.

💡 Research Summary

The paper investigates the energetic aspects of quantum correlations in continuous‑variable (CV) systems, focusing on Gaussian states that are central to optical quantum technologies. Traditional entanglement detection in CV platforms relies on information‑theoretic quantities such as the quantum mutual information, the positive‑partial‑transpose (PPT) criterion, or entropy‑based measures, which are often indirect and experimentally demanding. To provide a more operational and thermodynamically meaningful approach, the authors introduce an entropy‑free entanglement witness based on ergotropy—the maximal work extractable from a quantum state via unitary operations.

First, the authors define Gaussian ergotropy (E_G(\rho)) as the difference between the average energy of a state and that of its Gaussian passive counterpart, the latter being the state with the same symplectic spectrum but minimal energy under Gaussian unitaries. For a bipartite CV system, they distinguish between global ergotropy (obtained by joint Gaussian unitaries) and local ergotropy (obtained by product Gaussian unitaries acting on each mode separately). Their difference, the Gaussian ergotropic gap (\Delta E_G = E^{\text{global}}_G - E^{\text{local}}_G), quantifies the energetic advantage of correlated operations over local ones.

The authors prove that for pure two‑mode Gaussian states the gap vanishes if and only if the state is separable (Theorem 1). Moreover, (\Delta E_G) is a strictly increasing function of the quantum mutual information, establishing an equivalence between this energetic quantity and standard entanglement measures for pure states. However, for mixed states (\Delta E_G) merely signals the presence of correlations (Lemma 1) and can diverge with temperature, rendering it unsuitable as a quantitative entanglement indicator.

To overcome this limitation, the paper introduces the relative ergotropic gap (REG): \