- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

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

Experimentally Probing Entropy Reduction via Iterative Quantum Information Transfer

Toshihiro Yada1,*, Pieter-Jan Stas2,*, Aziza Suleymanzade2, Erik N. Knall3, Nobuyuki Yoshioka4,5, Takahiro Sagawa1,6, and Mikhail D. Lukin2

  • 1Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
  • 2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
  • 3John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
  • 4International Center for Elementary Particle Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
  • 5JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan
  • 6Quantum-Phase Electronics Center (QPEC), The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

  • *These authors contributed equally to this work.
PDF

Phys. Rev. X 15, 031054 – Published 26 August, 2025

DOI: https://doi.org/10.1103/5lp2-9sps

Abstract

Thermodynamic principles governing energy and information are important tools for a deeper understanding and better control of quantum systems. In this work, we experimentally investigate the interplay of the thermodynamic costs and information flow in a quantum system undergoing iterative quantum measurement and feedback. Our study employs a state stabilization protocol involving repeated measurement and feedback on an electronic spin qubit associated with a silicon-vacancy center in diamond, which is strongly coupled to a diamond nanocavity. This setup allows us to verify the fundamental laws of nonequilibrium quantum thermodynamics, including the second law and the fluctuation theorem, both of which incorporate measures of quantum information flow induced by iterative measurement and feedback. We further assess the reducible entropy based on the feedback’s causal structure and quantitatively demonstrate the thermodynamic advantages of non-Markovian feedback over Markovian feedback. For that purpose, we extend the theoretical framework of quantum thermodynamics to include the causal structure of the applied feedback protocol. Our work lays the foundation for investigating the entropic and energetic costs of real-time quantum control in various quantum systems.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Since the nineteenth-century thought experiment of Maxwell’s demon, scientists have explored how measurement and feedback affect thermodynamics. While theory has extended fundamental principles such as the second law and fluctuation theorems to include feedback, experimental studies of quantum information flow during real-time feedback control have remained elusive. In our work, we experimentally test how real-time quantum feedback reshapes thermodynamic behavior, offering new insights into the energetic and entropic costs of controlling quantum systems.

Our experimental platform is an electron spin qubit in a silicon-vacancy center embedded in a diamond nanocavity. By tracking how information flows from the qubit to the controller, we identify the fundamental limits of entropy reduction that feedback can achieve. We also explore the role of causal structure in the feedback loop, showing that non-Markovian (memory-assisted) feedback provides thermodynamic advantages over simpler Markovian feedback.

Our results build a crucial bridge between quantum thermodynamics and quantum information science. They open the door to designing more efficient quantum control schemes for tasks like quantum state preparation, stabilization, and error correction. Ultimately, this line of research could guide the development of high-performance quantum heat engines and batteries.

Article Text

Supplemental Material

References (76)

  1. S. Ciliberto, Experiments in stochastic thermodynamics: Short history and perspectives, Phys. Rev. X 7, 021051 (2017).
  2. M. Esposito, U. Harbola, and S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems, Rev. Mod. Phys. 81, 1665 (2009).
  3. M. Campisi, P. Hänggi, and P. Talkner, Colloquium: Quantum fluctuation relations: Foundations and applications, Rev. Mod. Phys. 83, 771 (2011).
  4. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  5. K. Funo, M. Ueda, and T. Sagawa, Quantum fluctuation theorems, in Thermodynamics in the Quantum Regime (Springer, New York, 2018), pp. 249–273.
  6. G. T. Landi and M. Paternostro, Irreversible entropy production: From classical to quantum, Rev. Mod. Phys. 93, 035008 (2021).
  7. H. Leff and A. F. Rex, Maxwell’s Demon 2 Entropy, Classical and Quantum Information, Computing (CRC Press, Boca Raton, FL, 2002).
  8. J. M. Parrondo, J. M. Horowitz, and T. Sagawa, Thermodynamics of information, Nat. Phys. 11, 131 (2015).
  9. T. Sagawa and M. Ueda, Second law of thermodynamics with discrete quantum feedback control, Phys. Rev. Lett. 100, 080403 (2008).
  10. T. Sagawa and M. Ueda, Generalized Jarzynski equality under nonequilibrium feedback control, Phys. Rev. Lett. 104, 090602 (2010).
  11. J. M. Horowitz and S. Vaikuntanathan, Nonequilibrium detailed fluctuation theorem for repeated discrete feedback, Phys. Rev. E 82, 061120 (2010).
  12. T. Sagawa and M. Ueda, Nonequilibrium thermodynamics of feedback control, Phys. Rev. E 85, 021104 (2012).
  13. S. Ito and T. Sagawa, Information thermodynamics on causal networks, Phys. Rev. Lett. 111, 180603 (2013).
  14. D. Hartich, A. C. Barato, and U. Seifert, Stochastic thermodynamics of bipartite systems: Transfer entropy inequalities and a Maxwell’s demon interpretation, J. Stat. Mech. (2014) P02016.
  15. J. M. Horowitz and M. Esposito, Thermodynamics with continuous information flow, Phys. Rev. X 4, 031015 (2014).
  16. K. Funo, Y. Watanabe, and M. Ueda, Integral quantum fluctuation theorems under measurement and feedback control, Phys. Rev. E 88, 052121 (2013).
  17. K. Ptaszyński and M. Esposito, Thermodynamics of quantum information flows, Phys. Rev. Lett. 122, 150603 (2019).
  18. T. Yada, N. Yoshioka, and T. Sagawa, Quantum fluctuation theorem under quantum jumps with continuous measurement and feedback, Phys. Rev. Lett. 128, 170601 (2022).
  19. P. Strasberg, G. Schaller, T. Brandes, and M. Esposito, Quantum and information thermodynamics: A unifying framework based on repeated interactions, Phys. Rev. X 7, 021003 (2017).
  20. P. Strasberg, Operational approach to quantum stochastic thermodynamics, Phys. Rev. E 100, 022127 (2019).
  21. M. Ribezzi-Crivellari and F. Ritort, Large work extraction and the Landauer limit in a continuous Maxwell demon, Nat. Phys. 15, 660 (2019).
  22. M. Debiossac, D. Grass, J. J. Alonso, E. Lutz, and N. Kiesel, Thermodynamics of continuous non-Markovian feedback control, Nat. Commun. 11, 1360 (2020).
  23. M. Debiossac, M. L. Rosinberg, E. Lutz, and N. Kiesel, Non-Markovian feedback control and acausality: An experimental study, Phys. Rev. Lett. 128, 200601 (2022).
  24. S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality, Nat. Phys. 6, 988 (2010).
  25. A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, Experimental verification of Landauer’s principle linking information and thermodynamics, Nature (London) 483, 187 (2012).
  26. É. Roldán, I. A. Martinez, J. M. Parrondo, and D. Petrov, Universal features in the energetics of symmetry breaking, Nat. Phys. 10, 457 (2014).
  27. J. V. Koski, V. F. Maisi, T. Sagawa, and J. P. Pekola, Experimental observation of the role of mutual information in the nonequilibrium dynamics of a Maxwell demon, Phys. Rev. Lett. 113, 030601 (2014).
  28. M. D. Vidrighin, O. Dahlsten, M. Barbieri, M. S. Kim, V. Vedral, and I. A. Walmsley, Photonic Maxwell’s demon, Phys. Rev. Lett. 116, 050401 (2016).
  29. K. Chida, S. Desai, K. Nishiguchi, and A. Fujiwara, Power generator driven by Maxwell’s demon, Nat. Commun. 8, 15301 (2017).
  30. G. Paneru, D. Y. Lee, T. Tlusty, and H. K. Pak, Lossless Brownian information engine, Phys. Rev. Lett. 120, 020601 (2018).
  31. G. Paneru, S. Dutta, T. Sagawa, T. Tlusty, and H. K. Pak, Efficiency fluctuations and noise induced refrigerator-to-heater transition in information engines, Nat. Commun. 11, 1012 (2020).
  32. L.-L. Yan, J.-T. Bu, Q. Zeng, K. Zhang, K.-F. Cui, F. Zhou, S.-L. Su, L. Chen, J. Wang, G. Chen, and M. Feng, Experimental verification of demon-involved fluctuation theorems, Phys. Rev. Lett. 133, 090402 (2024).
  33. P. A. Camati, J. P. S. Peterson, T. B. Batalhão, K. Micadei, A. M. Souza, R. S. Sarthour, I. S. Oliveira, and R. M. Serra, Experimental rectification of entropy production by Maxwell’s demon in a quantum system, Phys. Rev. Lett. 117, 240502 (2016).
  34. N. Cottet, S. Jezouin, L. Bretheau, P. Campagne-Ibarcq, Q. Ficheux, J. Anders, A. Auffèves, R. Azouit, P. Rouchon, and B. Huard, Observing a quantum Maxwell demon at work, Proc. Natl. Acad. Sci. U.S.A. 114, 7561 (2017).
  35. Y. Masuyama, K. Funo, Y. Murashita, A. Noguchi, S. Kono, Y. Tabuchi, R. Yamazaki, M. Ueda, and Y. Nakamura, Information-to-work conversion by Maxwell’s demon in a superconducting circuit quantum electrodynamical system, Nat. Commun. 9, 1291 (2018).
  36. M. Naghiloo, J. J. Alonso, A. Romito, E. Lutz, and K. W. Murch, Information gain and loss for a quantum Maxwell’s demon, Phys. Rev. Lett. 121, 030604 (2018).
  37. A. Solfanelli, A. Santini, and M. Campisi, Experimental verification of fluctuation relations with a quantum computer, PRX Quantum 2, 030353 (2021).
  38. S. Hernández-Gómez, S. Gherardini, N. Staudenmaier, F. Poggiali, M. Campisi, A. Trombettoni, F. S. Cataliotti, P. Cappellaro, and N. Fabbri, Autonomous dissipative Maxwell’s demon in a diamond spin qutrit, PRX Quantum 3, 020329 (2022).
  39. C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, S. Gleyzes, P. Rouchon, M. Mirrahimi, H. Amini, M. Brune, J.-M. Raimond, and S. Haroche, Real-time quantum feedback prepares and stabilizes photon number states, Nature (London) 477, 73 (2011).
  40. L. Magrini, P. Rosenzweig, C. Bach, A. Deutschmann-Olek, S. G. Hofer, S. Hong, N. Kiesel, A. Kugi, and M. Aspelmeyer, Real-time optimal quantum control of mechanical motion at room temperature, Nature (London) 595, 373 (2021).
  41. N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B. Vlastakis, Y. Liu, L. Frunzio, S. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, Extending the lifetime of a quantum bit with error correction in superconducting circuits, Nature (London) 536, 441 (2016).
  42. C. Ryan-Anderson, J. G. Bohnet, K. Lee, D. Gresh, A. Hankin, J. P. Gaebler, D. Francois, A. Chernoguzov, D. Lucchetti, N. C. Brown, T. M. Gatterman, S. K. Halit, K. Gilmore, J. A. Gerber, B. Neyenhuis, D. Hayes, and R. P. Stutz, Realization of real-time fault-tolerant quantum error correction, Phys. Rev. X 11, 041058 (2021).
  43. S. Krinner, N. Lacroix, A. Remm, A. Di Paolo, E. Genois, C. Leroux, C. Hellings, S. Lazar, F. Swiadek, J. Herrmann, G. J. Norris, C. K. Andersen, M. Müller, A. Blais, C. Eichler, and A. Wallraff, Realizing repeated quantum error correction in a distance-three surface code, Nature (London) 605, 669 (2022).
  44. H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, Cambridge, England, 2009).
  45. V. V. Sivak, A. Eickbusch, H. Liu, B. Royer, I. Tsioutsios, and M. H. Devoret, Model-free quantum control with reinforcement learning, Phys. Rev. X 12, 011059 (2022).
  46. J. M. Horowitz and H. Sandberg, Second-law-like inequalities with information and their interpretations, New J. Phys. 16, 125007 (2014).
  47. A. C. Barato, D. Hartich, and U. Seifert, Efficiency of cellular information processing, New J. Phys. 16, 103024 (2014).
  48. D. Hartich, A. C. Barato, and U. Seifert, Sensory capacity: An information theoretical measure of the performance of a sensor, Phys. Rev. E 93, 022116 (2016).
  49. S. Ito, Backward transfer entropy: Informational measure for detecting hidden Markov models and its interpretations in thermodynamics, gambling and causality, Sci. Rep. 6, 36831 (2016).
  50. C. T. Nguyen, D. D. Sukachev, M. K. Bhaskar, B. Machielse, D. S. Levonian, E. N. Knall, P. Stroganov, C. Chia, M. J. Burek, R. Riedinger, H. Park, M. Lončar, and M. D. Lukin, An integrated nanophotonic quantum register based on silicon-vacancy spins in diamond, Phys. Rev. B 100, 165428 (2019).
  51. P.-J. Stas, Y. Q. Huan, B. Machielse, E. N. Knall, A. Suleymanzade, B. Pingault, M. Sutula, S. W. Ding, C. M. Knaut, D. R. Assumpcao, Y.-C. Wei, M. K. Bhaskar, R. Riedinger, D. D. Sukachev, H. Park, M. Lončar, D. S. Levonian, and M. D. Lukin, Robust multi-qubit quantum network node with integrated error detection, Science 378, 557 (2022).
  52. S. Deffner and E. Lutz, Nonequilibrium entropy production for open quantum systems, Phys. Rev. Lett. 107, 140404 (2011).
  53. J. M. Horowitz and J. M. Parrondo, Entropy production along nonequilibrium quantum jump trajectories, New J. Phys. 15, 085028 (2013).
  54. F. W. J. Hekking and J. P. Pekola, Quantum jump approach for work and dissipation in a two-level system, Phys. Rev. Lett. 111, 093602 (2013).
  55. D. Andrieux and P. Gaspard, Fluctuation theorem and Onsager reciprocity relations, J. Chem. Phys. 121, 6167 (2004).
  56. K. Saito and Y. Utsumi, Symmetry in full counting statistics, fluctuation theorem, and relations among nonlinear transport coefficients in the presence of a magnetic field, Phys. Rev. B 78, 115429 (2008).
  57. T. Schreiber, Measuring information transfer, Phys. Rev. Lett. 85, 461 (2000).
  58. C. W. Granger, Investigating causal relations by econometric models and cross-spectral methods, Econometrica 37, 424 (1969).
  59. R. Vicente, M. Wibral, M. Lindner, and G. Pipa, Transfer entropy—A model-free measure of effective connectivity for the neurosciences, J. Comput. Neurosci. 30, 45 (2011).
  60. S. Ito and T. Sagawa, Maxwell’s demon in biochemical signal transduction with feedback loop, Nat. Commun. 6, 1 (2015).
  61. See Supplemental Material at http://link.aps.org/supplemental/10.1103/5lp2-9sps for detailed experimental procedures, additional experimental and numerical data, and the derivation of Eq. (6).
  62. P. D. Johnson, F. Ticozzi, and L. Viola, Exact stabilization of entangled states in finite time by dissipative quantum circuits, Phys. Rev. A 96, 012308 (2017).
  63. Y. Li and M. P. A. Fisher, Statistical mechanics of quantum error correcting codes, Phys. Rev. B 103, 104306 (2021).
  64. Google Quantum AI, Suppressing quantum errors by scaling a surface code logical qubit, Nature (London) 614, 676 (2023).
  65. H. Tajima and K. Funo, Superconducting-like heat current: Effective cancellation of current-dissipation trade-off by quantum coherence, Phys. Rev. Lett. 127, 190604 (2021).
  66. F. Campaioli, F. A. Pollock, F. C. Binder, L. Céleri, J. Goold, S. Vinjanampathy, and K. Modi, Enhancing the charging power of quantum batteries, Phys. Rev. Lett. 118, 150601 (2017).
  67. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, New York, 2002).
  68. G. Manzano and R. Zambrini, Quantum thermodynamics under continuous monitoring: A general framework, AVS Quantum Sci. 4, 025302 (2022).
  69. P. Strasberg and A. Winter, Stochastic thermodynamics with arbitrary interventions, Phys. Rev. E 100, 022135 (2019).
  70. P. Strasberg, Repeated interactions and quantum stochastic thermodynamics at strong coupling, Phys. Rev. Lett. 123, 180604 (2019).
  71. P. Strasberg, Thermodynamics of quantum causal models: An inclusive, Hamiltonian approach, Quantum 4, 240 (2020).
  72. P. Strasberg, Quantum Stochastic Thermodynamics: Foundations and Selected Applications (Oxford University Press, New York, 2022).
  73. A. S. Holevo, Bounds for the quantity of information transmitted by a quantum communication channel, Prob. Peredachi Inf. 9, 3 (1973), https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ppi&paperid=903&option_lang=eng.
  74. D. Petz, Monotonicity of quantum relative entropy revisited, Rev. Math. Phys. 15, 79 (2003).
  75. F. V. Jensen and T. D. Nielsen, Bayesian Networks and Decision Graphs (Springer, New York, 2007), Vol. 2.
  76. S. Ito, Information Thermodynamics on Causal Networks and Its Application to Biochemical Signal Transduction (Springer, New York, 2016).

Outline

Information

Phys. Rev. X 15, 031054– Published 26 August, 2025
  • Received 23 October 2024
  • Revised 16 May 2025
  • Accepted 17 July 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