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Ergotropic characterization of continuous variable entanglement

Beatriz Polo-Rodríguez, Federico Centrone, Gerardo Adesso, Mir Alimuddin

Abstract

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.

Ergotropic characterization of continuous variable entanglement

Abstract

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.

Paper Structure

This paper contains 15 sections, 7 theorems, 100 equations, 3 figures.

Table of Contents

  1. Appendix A: Separability of Gaussian two-mode states
  2. Appendix B: Proof of Theorem 1
  3. Appendix C: Parametrization of Gaussian two-mode mixed states
  4. Appendix D: Calculation of relative ergotropic gaps
  5. 1. Gaussian two-mode states & Proof of Lemma 1
  6. 2. REG and Quantum Discord in the Gaussian Regime
  7. Appendix E: Proof of Theorem 2 and Corollary 1
  8. Appendix F: Independency of relative ergotropic gap
  9. 1. Proof of Lemma 2
  10. 2. Comparison between entanglement witnesses
  11. 3. A total summary and the role of QMI and REG
  12. Appendix G: Non-Gaussian states
  13. 1. Symmetric superpositions of qudits
  14. 2. Mixtures of Bell states
  15. 3. Photon-subtracted TMS states

Key Result

Theorem 1

For pure two-mode Gaussian states $\rho$, the ergotropic gap $\Delta \mathcal{E} (\rho)$ vanishes if and only if $\rho$ is separable. Moreover, $\Delta \mathcal{E} (\rho)$ is a strictly increasing function of the quantum mutual information $I(\rho)$, and hence provides an equivalent quantification o

Figures (3)

  • Figure 1: Difference between $\mathcal{B}^{\text{sep}}_{\mathrm{max}}$ and $\Delta \mathcal{E}_{\text{rel}}$ for the parametric family of Gaussian two-mode squeezed states, as a function of the squeezing parameter $z$ and the mean fluctuations factor $k$. Values of fluctuation difference $\gamma$ have been set to {1,0.5,0} (left to right), and the frequency ratio is set at $\alpha=1$ for all cases. The dashed and solid red lines represent the boundaries between separable and entangled states according to our criterion and the PPT condition, respectively. As predicted by Corollary \ref{['corosame']}, these boundaries coincide when $\gamma = 0$.
  • Figure 2: G-REG of photon-subtracted TMS($z,k$) as a function of the fluctuation parameter $k$ of the initial thermal state and the Gaussian squeezing $z$ parameter. The solid line indicates the region of entangled states witnessed by the Shchukin-Vogel (SV) condition, namely those to the left of the line, and the dashed line indicates the entangled region as witnessed by our G-REG-based criterion.
  • Figure 3: REG for $N=5 \times 10^2$ random two-mode Gaussian states with fixed parameters $k=2.5$, $\gamma=0.5$ and $\alpha=10$. The blue and red dashed lines represent the lower and upper bounds for entangled and separable states, respectively.

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Corollary 1
  • Lemma 3
  • Lemma 4