- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Ergotropic Characterization of Continuous-Variable Entanglement

Beatriz Polo-Rodríguez1, Federico Centrone1,2,*, Gerardo Adesso3, and Mir Alimuddin1,†

  • *Contact author: federico.centrone@icfo.eu
  • †Contact author: aliphy80@gmail.com
PDF

Phys. Rev. Lett. 136, 050201 – Published 2 February, 2026

DOI: https://doi.org/10.1103/43jm-qkz2

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 Letter, 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.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (72)

  1. Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (Springer International Publishing, Cham, 2018).
  2. J. Goold, M. Huber, A. Riera, L. d. Rio, and P. Skrzypczyk, The role of quantum information in thermodynamics—A topical review, J. Phys. A 49, 143001 (2016).
  3. S. Vinjanampathy and J. Anders, Quantum thermodynamics, Contemp. Phys. 57, 545 (2016).
  4. M. Horodecki and J. Oppenheim, Fundamental limitations for quantum and nanoscale thermodynamics, Nat. Commun. 4, 2059 (2013).
  5. W. H. Zurek, Quantum discord and Maxwell’s demons, Phys. Rev. A 67, 012320 (2003).
  6. F. G. S. L. Brandão, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, Resource theory of quantum states out of thermal equilibrium, Phys. Rev. Lett. 111, 250404 (2013).
  7. P. Skrzypczyk, A. J. Short, and S. Popescu, Work extraction and thermodynamics for individual quantum systems, Nat. Commun. 5, 4185 (2014).
  8. S. Popescu, A. J. Short, and A. Winter, Entanglement and the foundations of statistical mechanics, Nat. Phys. 2, 754 (2006).
  9. G. Gour, M. P. Müller, V. Narasimhachar, R. W. Spekkens, and N. Yunger Halpern, The resource theory of informational nonequilibrium in thermodynamics, Phys. Rep. 583, 1 (2015).
  10. A. de Oliveira Junior, J. B. Brask, and P. Lipka-Bartosik, Heat as a witness of quantum properties, Phys. Rev. Lett. 134, 050401 (2025).
  11. N. Friis, M. Huber, and M. Perarnau-Llobet, Energetics of correlations in interacting systems, Phys. Rev. E 93, 042135 (2016).
  12. M. Huber, M. Perarnau-Llobet, K. V. Hovhannisyan, P. Skrzypczyk, C. Klöckl, N. Brunner, and A. Acín, Thermodynamic cost of creating correlations, New J. Phys. 17, 065008 (2015).
  13. G. Vitagliano, C. Klöckl, M. Huber, and N. Friis, Trade-off between work and correlations in quantum thermodynamics, in Thermodynamics in the Quantum Regime (Springer International Publishing, New York, 2018), pp. 731–750.
  14. A. Auffèves, Quantum technologies need a quantum energy initiative, PRX Quantum 3, 020101 (2022).
  15. A. E. Allahverdyan, R. Balian, and T. M. Nieuwenhuizen, Maximal work extraction from finite quantum systems, Europhys. Lett. 67, 565 (2004).
  16. G. Francica, F. C. Binder, G. Guarnieri, M. T. Mitchison, J. Goold, and F. Plastina, Quantum coherence and ergotropy, Phys. Rev. Lett. 125, 180603 (2020).
  17. B. Çakmak, Ergotropy from coherences in an open quantum system, Phys. Rev. E 102, 042111 (2020).
  18. R. Basu, A. Chakraborty, H. Badhani, M. Alimuddin, and S. Bhattacharya, Exploring non-Markovianity in ergodic channels: Measuring memory retention through ergotropy, Phys. Rev. A 111, 032416 (2025).
  19. D. Šafránek, D. Rosa, and F. C. Binder, Work extraction from unknown quantum sources, Phys. Rev. Lett. 130, 210401 (2023).
  20. G. Francica, J. Goold, F. Plastina, and M. Paternostro, Daemonic ergotropy: Enhanced work extraction from quantum correlations, npj Quantum Inf. 3, 12 (2017).
  21. A. Mukherjee, A. Roy, S. S. Bhattacharya, and M. Banik, Presence of quantum correlations results in a nonvanishing ergotropic gap, Phys. Rev. E 93, 052140 (2016).
  22. M. Alimuddin, T. Guha, and P. Parashar, Bound on ergotropic gap for bipartite separable states, Phys. Rev. A 99, 052320 (2019).
  23. M. Alimuddin, T. Guha, and P. Parashar, Independence of work and entropy for equal-energetic finite quantum systems: Passive-state energy as an entanglement quantifier, Phys. Rev. E 102, 012145 (2020).
  24. G. Francica, Quantum correlations and ergotropy, Phys. Rev. E 105, L052101 (2022).
  25. J. Joshi, M. Alimuddin, T. S. Mahesh, and M. Banik, Experimental verification of many-body entanglement using thermodynamic quantities, Phys. Rev. A 109, L020403 (2024).
  26. S. Puliyil, M. Banik, and M. Alimuddin, Thermodynamic signatures of genuinely multipartite entanglement, Phys. Rev. Lett. 129, 070601 (2022).
  27. X. Yang, Y.-H. Yang, M. Alimuddin, R. Salvia, S.-M. Fei, L.-M. Zhao, S. Nimmrichter, and M.-X. Luo, Battery capacity of energy-storing quantum systems, Phys. Rev. Lett. 131, 030402 (2023).
  28. A. Sone and S. Deffner, Quantum and classical ergotropy from relative entropies, Entropy 23, 1107 (2021).
  29. R. P. A. Simon, J. Anders, and K. V. Hovhannisyan, Correlations enable lossless ergotropy transport, Phys. Rev. Lett. 134, 010408 (2025).
  30. X. Yang, M. Alimuddin, Y.-H. Yang, and M.-X. Luo, Characterizing bipartite entanglement via the ergotropic gap, Quantum Inf. Process. 23, 382 (2024).
  31. M. Alimuddin, S. R. Chowdhury, R. K. Patra, S. B. Ghosh, T. Tufarelli, G. Adesso, and M. Banik, Ergodiscord: An operational and distinct notion of quantumness of correlations, Phys. Rev. Res. 7, 023200 (2025).
  32. S. L. Braunstein and P. van Loock, Quantum information with continuous variables, Rev. Mod. Phys. 77, 513 (2005).
  33. A. Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods (CRC Press, Boca Raton, 2023).
  34. G. Adesso, S. Ragy, and A. R. Lee, Continuous variable quantum information: Gaussian states and beyond, Open Syst. Inf. Dyn. 21, 1440001 (2014).
  35. C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012).
  36. G. Adesso, A. Serafini, and F. Illuminati, Extremal entanglement and mixedness in continuous variable systems, Phys. Rev. A 70, 022318 (2004).
  37. L. Lami, A. Serafini, and G. Adesso, Gaussian entanglement revisited, New J. Phys. 20, 023030 (2018).
  38. Y. Huang, Entanglement detection: Complexity and Shannon entropic criteria, IEEE Trans. Inf. Theory 59, 6774 (2013).
  39. M. Perarnau-Llobet, K. V. Hovhannisyan, M. Huber, P. Skrzypczyk, N. Brunner, and A. Acín, Extractable work from correlations, Phys. Rev. X 5, 041011 (2015).
  40. F. Centrone, L. Mancino, and M. Paternostro, Charging batteries with quantum squeezing, Phys. Rev. A 108, 052213 (2023).
  41. M. Walschaers, Non-Gaussian quantum states and where to find them, PRX Quantum 2, 030204 (2021).
  42. E. Shchukin and W. Vogel, Inseparability criteria for continuous bipartite quantum states, Phys. Rev. Lett. 95, 230502 (2005).
  43. W. Pusz and S. L. Woronowicz, Passive states and KMS states for general quantum systems, Commun. Math. Phys. 58, 273 (1978).
  44. A. Lenard, Thermodynamical proof of the Gibbs formula for elementary quantum systems, J. Stat. Phys. 19, 575 (1978).
  45. P. Skrzypczyk, R. Silva, and N. Brunner, Passivity, complete passivity, and virtual temperatures, Phys. Rev. E 91, 052133 (2015).
  46. E. G. Brown, N. Friis, and M. Huber, Passivity and practical work extraction using Gaussian operations, New J. Phys. 18, 113028 (2016).
  47. F. E. Q. Rodriguez, F. A. Mele, S. F. E. Oliviero, V. Giovannetti, L. Lami, and V. Cavina, Extracting energy via bosonic Gaussian operations, arXiv:2503.21748.
  48. A. Serafini, M. Lostaglio, S. Longden, U. Shackerley-Bennett, C.-Y. Hsieh, and G. Adesso, Gaussian thermal operations and the limits of algorithmic cooling, Phys. Rev. Lett. 124, 010602 (2020).
  49. See Supplemental Material at http://link.aps.org/supplemental/10.1103/43jm-qkz2 for detailed calculations, which includes Refs. [50–60].
  50. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).
  51. M. Horodecki, P. Horodecki, and R. Horodecki, Separability of mixed states: Necessary and sufficient conditions, Phys. Lett. A 223, 1 (1996).
  52. M. J. Donald, M. Horodecki, and O. Rudolph, The uniqueness theorem for entanglement measures, J. Math. Phys. (N.Y.) 43, 4252–4272 (2002).
  53. W. H. Zurek, Einselection and decoherence from an information theory perspective, Ann. Phys. (N.Y.) 512, 855 (2000).
  54. H. Ollivier and W. H. Zurek, Quantum discord: A measure of the quantumness of correlations, Phys. Rev. Lett. 88, 017901 (2001).
  55. L. Henderson and V. Vedral, Classical, quantum and total correlations, J. Phys. A 34, 6899 (2001).
  56. G. Adesso and A. Datta, Quantum versus classical correlations in Gaussian states, Phys. Rev. Lett. 105, 030501 (2010).
  57. R. Horodecki and P. Horodecki, Quantum redundancies and local realism, Phys. Lett. A 194, 147 (1994).
  58. A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50, 221 (1978).
  59. M. M. Wilde, Preface to the second edition, in Quantum Information Theory (Cambridge University Press, Cambridge, England, 2017), pp. xi–xii.
  60. R. Horodecki and M. Horodecki, Information-theoretic aspects of inseparability of mixed states, Phys. Rev. A 54, 1838 (1996).
  61. G. Vidal, Entanglement monotones, J. Mod. Opt. 47, 355 (2000).
  62. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2000).
  63. A. Botero and B. Reznik, Modewise entanglement of Gaussian states, Phys. Rev. A 67, 052311 (2003).
  64. D. P. DiVincenzo, M. Horodecki, D. W. Leung, J. A. Smolin, and B. M. Terhal, Locking classical correlations in quantum states, Phys. Rev. Lett. 92, 067902 (2004).
  65. S. L. Braunstein, Squeezing as an irreducible resource, Phys. Rev. A 71, 055801 (2005).
  66. R. Simon, Peres-Horodecki separability criterion for continuous variable systems, Phys. Rev. Lett. 84, 2726 (2000).
  67. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  68. G. Adesso, A. Serafini, and F. Illuminati, Determination of continuous variable entanglement by purity measurements, Phys. Rev. Lett. 92, 087901 (2004).
  69. L. Lami, B. Regula, X. Wang, R. Nichols, A. Winter, and G. Adesso, Gaussian quantum resource theories, Phys. Rev. A 98, 022335 (2018).
  70. M. Walschaers, C. Fabre, V. Parigi, and N. Treps, Entanglement and Wigner function negativity of multimode non-Gaussian states, Phys. Rev. Lett. 119, 183601 (2017).
  71. I. de Buy Wenniger, S. Thomas, M. Maffei, S. Wein, M. Pont, N. Belabas, S. Prasad, A. Harouri, A. Lema𝚤tre, I. Sagnes et al., Experimental analysis of energy transfers between a quantum emitter and light fields, Phys. Rev. Lett. 131, 260401 (2023).
  72. B. Polo-Rodríguez, F. Centrone, G. Adesso, and M. Alimuddin, Dataset for “Ergotropic characterization of continuous variable entanglement”, 2025, GitHub repository, https://github.com/beapolo13/CVEntanglement.

Outline

Information

Phys. Rev. Lett. 136, 050201– Published 2 February, 2026
  • Received 11 June 2025
  • Accepted 9 December 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation