The generalized quantum Stein’s lemma and the second law of quantum resource theories

Nature Physics (2025)Cite this article The second law of thermodynamics is a fundamental concept in physics, characterizing the convertibility between thermodynamic states through a single function—entropy. An important question in quantum information theory has been whether an analogous second law can be established for resources in quantum information processing, such as entanglement. In 2008, a formulation was proposed, linking resource convertibility to the optimal performance of a variant of the quantum version of hypothesis testing. The proposal made use of the generalized quantum Stein’s lemma to characterize this optimal performance by a measure of quantum resources, the regularized relative entropy of resource. If this approach is valid, a second law for quantum resources can be established, with the regularized relative entropy of resource taking on the role of thermodynamic entropy. However, in 2023, a gap was found in the proof of the generalized Stein’s lemma. Here we provide an alternative proof of the generalized quantum Stein’s lemma under a smaller set of assumptions. Furthermore, we re-establish and extend the second law of quantum resource theories, applicable to both static resources of quantum states and dynamical resources represented by classical–quantum channels.This is a preview of subscription content, access via your institution Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $32.99 / 30 days cancel any timeSubscribe to this journal Receive 12 print issues and online access $259.00 per yearonly $21.58 per issueBuy this articlePrices may be subject to local taxes which are calculated during checkoutNielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge Univ. Press, 2011)Kuroiwa, K. & Yamasaki, H. General quantum resource theories: distillation, formation and consistent resource measures. Quantum 4, 355 (2020).Article Google Scholar Chitambar, E. & Gour, G. Quantum resource theories. Rev. Mod. Phys. 91, 025001 (2019).Article ADS MathSciNet Google Scholar Horodecki, R., Horodecki, P., Horodecki, M. & Horodecki, K. Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009).Article ADS MathSciNet MATH Google Scholar Chitambar, E., Leung, D., Mančinska, L., Ozols, M. & Winter, A. Everything you always wanted to know about LOCC (but were afraid to ask). Commun. Math. Phys. 328, 303 (2014).Article ADS MathSciNet MATH Google Scholar Yamasaki, H., Kuroiwa, K., Hayden, P. & Lami, L. Entanglement cost for infinite-dimensional physical systems. Preprint at https://arxiv.org/abs/2401.09554 (2024).Carnot, S. Reflections on the Motive Power of Heat and on Machines Fitted to Develop that Power (J. Wiley, 1890).Clausius, R. On a modified form of the second fundamental theorem in the mechanical theory of heat. London Edinb. Dublin Philos. Mag. J. Sci. 12, 81 (1856).Thomson, W. On the dynamical theory of heat, with numerical results deduced from Mr Joule’s equivalent of a thermal unit, and M. Regnault’s observations on steam. Trans. R. Soc. Edinb. 20, 261 (1853).Article Google Scholar Lieb, E. H. & Yngvason, J. The physics and mathematics of the second law of thermodynamics. Phys. Rep. 310, 1 (1999).Article ADS MathSciNet Google Scholar Lieb, E. H. & Yngvason, J. in Statistical Mechanics: Selecta of Elliott H. Lieb (eds Nachtergaele, B., Solovej, J. P. & Yngvason, J.) 353–363 (Springer, 2004).Lieb, E. H. & Yngvason, J. A fresh look at entropy and the second law of thermodynamics. Phys. Today 53, 32 (2000).Article Google Scholar Lewis, G. & Randall, M. Thermodynamics and the Free Energy of Chemical Substances (McGraw-Hill, 1923).Guggenheim, E. Modern Thermodynamics by the Methods of Willard Gibbs (Methuen & Company Limited, 1933).Landauer, R. Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5, 183 (1961).Article MathSciNet MATH Google Scholar Meier, F. & Yamasaki, H. Energy-consumption advantage of quantum computation. PRX Energy 4, 023008 (2025).Article Google Scholar Shannon, C. E. A mathematical theory of communication. Bell Syst. Tech. J. 27, 379 (1948).Article ADS MathSciNet Google Scholar Cover, T. & Thomas, J. Elements of Information Theory (Wiley, 2012).Vidal, G. & Cirac, J. I. Irreversibility in asymptotic manipulations of entanglement. Phys. Rev. Lett. 86, 5803 (2001).Article ADS Google Scholar Wang, X. & Duan, R. Irreversibility of asymptotic entanglement manipulation under quantum operations completely preserving positivity of partial transpose. Phys. Rev. Lett. 119, 180506 (2017).Article ADS Google Scholar Lami, L. & Regula, B. No second law of entanglement manipulation after all. Nat. Phys. 19, 184 (2023).

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H.Y. was supported by JST PRESTO grant number JPMJPR201A, JPMJPR23FC, JSPS KAKENHI grant number JP23K19970, and MEXT Quantum Leap Flagship Program (MEXT QLEAP) JPMXS0118069605 and JPMXS0120351339.School of Data Science, The Chinese University of Hong Kong, Shenzhen, Shenzhen, ChinaMasahito HayashiInternational Quantum Academy, Shenzhen, ChinaMasahito HayashiGraduate School of Mathematics, Nagoya University, Nagoya, JapanMasahito HayashiDepartment of Physics, Graduate School of Science, The University of Tokyo, Bunkyo-ku, JapanHayata YamasakiDepartment of Computer Science, Graduate School of Information Science and Technology, The University of Tokyo, Bunkyo-ku, JapanHayata YamasakiSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarH.Y. and M.H. contributed to the conception of the work, the analysis and interpretation of the work and the preparation of the manuscript.Correspondence to Masahito Hayashi or Hayata Yamasaki.The authors declare no competing interests.Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Supplementary Sections I–III.Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Reprints and permissionsHayashi, M., Yamasaki, H. The generalized quantum Stein’s lemma and the second law of quantum resource theories. Nat. Phys. (2025). https://doi.org/10.1038/s41567-025-03047-9Download citationReceived: 20 October 2024Accepted: 27 August 2025Published: 29 October 2025DOI: https://doi.org/10.1038/s41567-025-03047-9Anyone you share the following link with will be able to read this content:Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative