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Quantum Geometric Tensor Determines the Pure-State I.I.D. Conversion Rate in the Resource Theory of Asymmetry for Any Compact Lie Group

Koji Yamaguchi1,5,*, Yosuke Mitsuhashi2,6, Tomohiro Shitara3, and Hiroyasu Tajima1,4,5

  • *Present address: Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.
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Phys. Rev. X 16, 031028 – Published 5 August, 2026

DOI: https://doi.org/10.1103/qqf6-x85b

Abstract

Quantifying physical concepts in terms of the ultimate performance of a given task has been central to theoretical progress, as illustrated by thermodynamic entropy and entanglement entropy, which quantify irreversibility and quantum correlations, respectively. Symmetry breaking is equally universal yet lacks such an operational quantification. While an operational characterization of symmetry breaking through asymptotic state-conversion efficiency is a central goal of the resource theory of asymmetry (RTA), such a characterization has so far been completed only for the U(1) group among continuous symmetries. Here, we identify the complete measure of symmetry breaking for a general continuous symmetry described by any compact Lie group. Specifically, we show that the asymptotic conversion rate between many copies of pure states in RTA is determined by the quantum geometric tensor, thereby establishing it as the complete measure of symmetry breaking. As an immediate consequence of our conversion rate formula, we also resolve the Marvian-Spekkens conjecture on conditions for reversible conversion in RTA, which has remained unproven for over a decade. Leveraging the connection between symmetry breaking and the theory of quantum reference frames, we also systematically introduce a standardized reference state for frameness based on our asymptotic conversion theory. In addition, by applying our analysis to a standard quantum-thermodynamic scenario, we show that asymptotic state conversion in contact with heat baths generally requires macroscopic coherence in the thermodynamic limit.

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Popular Summary

Symmetry plays a central role in physics, yet quantifying symmetry breaking remains a major challenge. We solve a decade-old open problem by identifying a complete characterization of pure-state asymptotic convertibility under continuous symmetries. Specifically, we show that the optimal conversion rate is fully determined by the quantum geometric tensor. This establishes this differential-geometric object as the fundamental measure of symmetry breaking, providing information-theoretic foundations analogous to entanglement entropy in entanglement theory. As direct applications, our framework introduces a benchmark for quantum reference frames, such as quantum clocks and gyroscopes, and demonstrates that state conversion under thermal contact generally requires macroscopic quantum coherence. More broadly, our results provide a common language for studying symmetry breaking across quantum information and condensed-matter physics, revealing a conceptual shift: the usefulness of quantum states for quantum tasks cannot always be captured by a scalar-valued function, but instead by an intrinsically operator-valued function.

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References (140)

  1. E. H. Lieb and J. Yngvason, The physics and mathematics of the second law of thermodynamics, Phys. Rep. 310, 1 (1999).
  2. C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Concentrating partial entanglement by local operations, Phys. Rev. A 53, 2046 (1996).
  3. G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Entanglement in quantum critical phenomena, Phys. Rev. Lett. 90, 227902 (2003).
  4. P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002.
  5. S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from the anti–de Sitter space/conformal field theory correspondence, Phys. Rev. Lett. 96, 181602 (2006).
  6. E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys. 91, 025001 (2019).
  7. T. Sagawa, Entropy, Divergence, and Majorization in Classical and Quantum Thermodynamics, SpringerBriefs in Mathematical Physics Vol. 16 (Springer, Singapore, 2022).
  8. C. Datta, R. Ganardi, T. V. Kondra, and A. Streltsov, Is there a finite complete set of monotones in any quantum resource theory?, Phys. Rev. Lett. 130, 240204 (2023).
  9. M. Horodecki, P. Horodecki, and J. Oppenheim, Reversible transformations from pure to mixed states and the unique measure of information, Phys. Rev. A 67, 062104 (2003).
  10. 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).
  11. S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Reference frames, superselection rules, and quantum information, Rev. Mod. Phys. 79, 555 (2007).
  12. G. Gour and R. W. Spekkens, The resource theory of quantum reference frames: Manipulations and monotones, New J. Phys. 10, 033023 (2008).
  13. G. Gour, I. Marvian, and R. W. Spekkens, Measuring the quality of a quantum reference frame: The relative entropy of frameness, Phys. Rev. A 80, 012307 (2009).
  14. I. Marvian Mashhad, Symmetry, asymmetry and quantum information, Ph.D. thesis, University of Waterloo, 2012.
  15. K. Korzekwa, Resource theory of asymmetry, Master’s thesis, Imperial College London, 2013.
  16. I. Marvian and R. W. Spekkens, Asymmetry properties of pure quantum states, Phys. Rev. A 90, 014102 (2014).
  17. I. Marvian, R. W. Spekkens, and P. Zanardi, Quantum speed limits, coherence, and asymmetry, Phys. Rev. A 93, 052331 (2016).
  18. I. Marvian and R. W. Spekkens, No-broadcasting theorem for quantum asymmetry and coherence and a trade-off relation for approximate broadcasting, Phys. Rev. Lett. 123, 020404 (2019).
  19. M. Lostaglio and M. P. Müller, Coherence and asymmetry cannot be broadcast, Phys. Rev. Lett. 123, 020403 (2019).
  20. M. Lostaglio, K. Korzekwa, D. Jennings, and T. Rudolph, Quantum coherence, time-translation symmetry, and thermodynamics, Phys. Rev. X 5, 021001 (2015).
  21. P. Faist, J. Oppenheim, and R. Renner, Gibbs-preserving maps outperform thermal operations in the quantum regime, New J. Phys. 17, 043003 (2015).
  22. H. Tajima and R. Takagi, Gibbs-preserving operations requiring infinite amount of quantum coherence, Phys. Rev. Lett. 134, 170201 (2025).
  23. I. Marvian, Coherence distillation machines are impossible in quantum thermodynamics, Nat. Commun. 11, 25 (2020).
  24. I. Marvian, Operational interpretation of quantum Fisher information in quantum thermodynamics, Phys. Rev. Lett. 129, 190502 (2022).
  25. A. Kubica and R. Demkowicz-Dobrzański, Using quantum metrological bounds in quantum error correction: A simple proof of the approximate Eastin-Knill theorem, Phys. Rev. Lett. 126, 150503 (2021).
  26. S. Zhou, Z.-W. Liu, and L. Jiang, New perspectives on covariant quantum error correction, Quantum 5, 521 (2021).
  27. Y. Yang, Y. Mo, J. M. Renes, G. Chiribella, and M. P. Woods, Optimal universal quantum error correction via bounded reference frames, Phys. Rev. Res. 4, 023107 (2022).
  28. H. Tajima and K. Saito, Universal limitation of quantum information recovery: Symmetry versus coherence, arXiv:2103.01876.
  29. Z.-W. Liu and S. Zhou, Quantum error correction meets continuous symmetries: Fundamental trade-offs and case studies, arXiv:2111.06360.
  30. H. Tajima, R. Takagi, and Y. Kuramochi, Universal trade-off structure between symmetry, irreversibility, and quantum coherence in quantum processes, arXiv:2206.11086.
  31. Z.-W. Liu and S. Zhou, Approximate symmetries and quantum error correction, npj Quantum Inf. 9, 1 (2023).
  32. H. Tajima, N. Shiraishi, and K. Saito, Uncertainty relations in implementation of unitary operations, Phys. Rev. Lett. 121, 110403 (2018).
  33. H. Tajima, N. Shiraishi, and K. Saito, Coherence cost for violating conservation laws, Phys. Rev. Res. 2, 043374 (2020).
  34. M. Ahmadi, D. Jennings, and T. Rudolph, The way theorem and the quantum resource theory of asymmetry, New J. Phys. 15, 013057 (2013).
  35. I. Marvian and R. W. Spekkens, An information-theoretic account of the Wigner-Araki-Yanase theorem, arXiv:1212.3378.
  36. H. Tajima and H. Nagaoka, Coherence-variance uncertainty relation and coherence cost for quantum measurement under conservation laws, arXiv:1909.02904.
  37. R. Takagi, Skew informations from an operational view via resource theory of asymmetry, Sci. Rep. 9, 14562 (2019).
  38. Y. Yang, G. Chiribella, and Q. Hu, Units of rotational information, New J. Phys. 19, 123003 (2017).
  39. H. Emori and H. Tajima, Error and disturbance as irreversibility with applications: Unified definition, Wigner–Araki–Vanase theorem and out-of-time-order correlator, arXiv:2309.14172.
  40. F. D. M. Haldane, Nonlinear field theory of large-spin Heisenberg antiferromagnets: Semiclassically quantized solitons of the one-dimensional easy-axis Néel state, Phys. Rev. Lett. 50, 1153 (1983).
  41. I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets, Phys. Rev. Lett. 59, 799 (1987).
  42. D. Jaksch and P. Zoller, The cold atom Hubbard toolbox, Ann. Phys. (Amsterdam) 315, 52 (2005).
  43. C. R. Dean, A. F. Young, P. Cadden-Zimansky, L. Wang, H. Ren, K. Watanabe, T. Taniguchi, P. Kim, J. Hone, and K. L. Shepard, Multicomponent fractional quantum Hall effect in graphene, Nat. Phys. 7, 693 (2011).
  44. J. P. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Commun. Math. Phys. 76, 289 (1980).
  45. M. V. Berry, The quantum phase, five years after, in Geometric Phases in Physics (World Scientific, Singapore, 1989), Vol. 5, pp. 3–28.
  46. D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49, 405 (1982).
  47. J.-H. Zhao and H.-Q. Zhou, Singularities in ground-state fidelity and quantum phase transitions for the Kitaev model, Phys. Rev. B 80, 014403 (2009).
  48. S.-J. Gu, Fidelity approach to quantum phase transitions, Int. J. Mod. Phys. B 24, 4371 (2010).
  49. D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
  50. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, England, 2010).
  51. A. K. Ekert, Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett. 67, 661 (1991).
  52. J. I. Cirac, A. K. Ekert, S. F. Huelga, and C. Macchiavello, Distributed quantum computation over noisy channels, Phys. Rev. A 59, 4249 (1999).
  53. H. J. Kimble, The quantum internet, Nature (London) 453, 1023 (2008).
  54. M. Lostaglio, D. Jennings, and T. Rudolph, Description of quantum coherence in thermodynamic processes requires constraints beyond free energy, Nat. Commun. 6, 6383 (2015).
  55. M. Weilenmann, L. Kraemer, P. Faist, and R. Renner, Axiomatic relation between thermodynamic and information-theoretic entropies, Phys. Rev. Lett. 117, 260601 (2016).
  56. 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).
  57. P. Faist, T. Sagawa, K. Kato, H. Nagaoka, and F. G. Brandão, Macroscopic thermodynamic reversibility in quantum many-body systems, Phys. Rev. Lett. 123, 250601 (2019).
  58. T. Sagawa, P. Faist, K. Kato, K. Matsumoto, H. Nagaoka, and F. G. S. L. Brandão, Asymptotic reversibility of thermal operations for interacting quantum spin systems via generalized quantum Stein’s lemma, J. Phys. A 54, 495303 (2021).
  59. P. Lipka-Bartosik and P. Skrzypczyk, All states are universal catalysts in quantum thermodynamics, Phys. Rev. X 11, 011061 (2021).
  60. M. P. Woods and M. Horodecki, Autonomous quantum devices: When are they realizable without additional thermodynamic costs?, Phys. Rev. X 13, 011016 (2023).
  61. T. V. Kondra, R. Ganardi, and A. Streltsov, Coherence manipulation in asymmetry and thermodynamics, Phys. Rev. Lett. 132, 200201 (2024).
  62. N. Shiraishi, Quantum thermodynamics with coherence: Covariant Gibbs-preserving operation is characterized by the free energy, Phys. Rev. Lett. 134, 160402 (2025).
  63. G. Zambon and G. Adesso, Quantum processes as thermodynamic resources: The role of non-Markovianity, Phys. Rev. Lett. 134, 200401 (2025).
  64. T. Shitara, Y. Mitsuhashi, and H. Tajima, The i.i.d. state convertibility in the resource theory of asymmetry for finite groups (2024), arXiv:2312.15758.
  65. E. A. Morozova and N. N. Chentsov, Markov invariant geometry on state manifolds (in Russian), Itogi Nauk. Tekh. 36, 69 (1989).
  66. D. Petz, Monotone metrics on matrix spaces, Linear Algebra Appl. 244, 81 (1996).
  67. M. J. Donald, M. Horodecki, and O. Rudolph, The uniqueness theorem for entanglement measures, J. Math. Phys. (N.Y.) 43, 4252 (2002).
  68. M. Plenio and S. Virmani, An introduction to entanglement measures, Quantum Inf. Comput. 7, 1 (2007).
  69. K. Yamaguchi and H. Tajima, Smooth metric adjusted skew information rates, Quantum 7, 1012 (2023).
  70. M. Guţă and J. Kahn, Local asymptotic normality for qubit states, Phys. Rev. A 73, 052108 (2006).
  71. J. Kahn and M. Guţă, Local asymptotic normality for finite dimensional quantum systems, Commun. Math. Phys. 289, 597 (2009).
  72. F. Girotti, A. Godley, and M. Guţă, Optimal estimation of pure states with displaced-null measurements, J. Phys. A 57, 245304 (2024).
  73. S. Lahiry and M. Nussbaum, Minimax estimation of low-rank quantum states and their linear functionals, Bernoulli 30, 610 (2024).
  74. P. M. Hayden, M. Horodecki, and B. M. Terhal, The asymptotic entanglement cost of preparing a quantum state, J. Phys. A 34, 6891 (2001).
  75. M. Keyl and R. F. Werner, Optimal cloning of pure states, testing single clones, J. Math. Phys. (N.Y.) 40, 3283 (1999).
  76. C. W. Helstrom, Quantum detection and estimation theory, J. Stat. Phys. 1, 231 (1969).
  77. A. S. Holevo, Statistical decision theory for quantum systems, J. Multivariate Anal. 3, 337 (1973).
  78. R. Cheng, Quantum geometric tensor (Fubini-Study metric) in simple quantum system: A pedagogical introduction, arXiv:1012.1337.
  79. J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum Fisher information matrix and multiparameter estimation, J. Phys. A 53, 023001 (2019).
  80. L. Gao, H. Li, I. Marvian, and C. Rouzé, Sufficient statistic and recoverability via quantum Fisher information, Commun. Math. Phys. 405, 180 (2024).
  81. D. Kudo and H. Tajima, Fisher information matrix as a resource measure in the resource theory of asymmetry with general connected-Lie-group symmetry, Phys. Rev. A 107, 062418 (2023).
  82. N. Datta, Min- and max-relative entropies and a new entanglement monotone, IEEE Trans. Inf. Theory 55, 2816 (2009).
  83. M. Tomamichel, Quantum Information Processing with Finite Resources, SpringerBriefs in Mathematical Physics Vol. 5 (Springer International Publishing, Cham, 2016).
  84. N. Datta, Max-relative entropy of entanglement, alias log robustness, Int. J. Quantum. Inform. 07, 475 (2009).
  85. M. Tomamichel, R. Colbeck, and R. Renner, A fully quantum asymptotic equipartition property, IEEE Trans. Inf. Theory 55, 5840 (2009).
  86. R. Konig, R. Renner, and C. Schaffner, The operational meaning of min- and max-entropy, IEEE Trans. Inf. Theory 55, 4337 (2009).
  87. M. Tomamichel and R. Renner, Uncertainty relation for smooth entropies, Phys. Rev. Lett. 106, 110506 (2011).
  88. K. Bu, U. Singh, S.-M. Fei, A. K. Pati, and J. Wu, Maximum relative entropy of coherence: An operational coherence measure, Phys. Rev. Lett. 119, 150405 (2017).
  89. N. Schuch, F. Verstraete, and J. I. Cirac, Nonlocal resources in the presence of superselection rules, Phys. Rev. Lett. 92, 087904 (2004).
  90. N. Schuch, F. Verstraete, and J. I. Cirac, Quantum entanglement theory in the presence of superselection rules, Phys. Rev. A 70, 042310 (2004).
  91. K. Yamaguchi and H. Tajima, Beyond i.i.d. in the resource theory of asymmetry: An information-spectrum approach for quantum Fisher information, Phys. Rev. Lett. 131, 200203 (2023).
  92. I. Marvian and R. W. Spekkens, The theory of manipulations of pure state asymmetry: I. Basic tools, equivalence classes and single copy transformations, New J. Phys. 15, 033001 (2013).

  93. Formally, a finite group can be regarded as a Lie group, since it can be viewed as a zero-dimensional smooth manifold. However, in our terminology, we consider Lie groups to have a dimension greater than zero.

  94. D. Petz and C. Ghinea, Introduction to quantum fisher information, in Quantum Probability and Related Topics, QP-PQ: Quantum Probability and White Noise Analysis Vol. 27 (World Scientific, Singapore, 2011), pp. 261–281.
  95. F. Hansen, Metric adjusted skew information, Proc. Natl. Acad. Sci. U.S.A. 105, 9909 (2008).

  96. We remark that not all symmetric mixed states can be decomposed as an ensemble of symmetric pure states. Indeed, when a unitary representation of a group is decomposed into irreducible components, Schur’s lemma implies that only a one-dimensional irreducible representation contains a symmetric pure state. In other words, no symmetric pure state exists in any higher-dimensional irreducible representation.

  97. G. Tóth and D. Petz, Extremal properties of the variance and the quantum Fisher information, Phys. Rev. A 87, 032324 (2013).
  98. S. Yu, Quantum Fisher information as the convex roof of variance, arXiv:1302.5311.
  99. Y. Aharonov and L. Susskind, Charge superselection rule, Phys. Rev. 155, 1428 (1967).
  100. M. R. Dowling, S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Observing a coherent superposition of an atom and a molecule, Phys. Rev. A 74, 052113 (2006).
  101. F. Giacomini, E. Castro-Ruiz, and v. Brukner, Quantum mechanics and the covariance of physical laws in quantum reference frames, Nat. Commun. 10, 494 (2019).
  102. A. Vanrietvelde, P. A. Hoehn, F. Giacomini, and E. Castro-Ruiz, A change of perspective: Switching quantum reference frames via a perspective-neutral framework, Quantum 4, 225 (2020).
  103. P. A. Höhn, A. R. H. Smith, and M. P. E. Lock, Trinity of relational quantum dynamics, Phys. Rev. D 104, 066001 (2021).
  104. V. Chandrasekaran, R. Longo, G. Penington, and E. Witten, An algebra of observables for de Sitter space, J. High Energy Phys. 02 (2023) 082.
  105. C. J. Fewster, D. W. Janssen, L. D. Loveridge, K. Rejzner, and J. Waldron, Quantum reference frames, measurement schemes and the type of local algebras in quantum field theory, Commun. Math. Phys. 406, 19 (2024).
  106. J. De Vuyst, S. Eccles, P. A. Höhn, and J. Kirklin, Crossed products and quantum reference frames: On the observer-dependence of gravitational entropy, J. High Energy Phys. 07 (2025) 063.
  107. I. Marvian and R. W. Spekkens, How to quantify coherence: Distinguishing speakable and unspeakable notions, Phys. Rev. A 94, 052324 (2016).
  108. A. Peres and P. F. Scudo, Unspeakable quantum information, in Quantum Theory: Reconsideration of Foundations-2 (Vaxjo University Press, Vaxjo, Sweden, 2002).
  109. H. Georgi, Lie Algebras In Particle Physics: From Isospin To Unified Theories (CRC Press, Boca Raton, 2018).
  110. C. Zhang, B. Yadin, Z.-B. Hou, H. Cao, B.-H. Liu, Y.-F. Huang, R. Maity, V. Vedral, C.-F. Li, G.-C. Guo, and D. Girolami, Detecting metrologically useful asymmetry and entanglement by a few local measurements, Phys. Rev. A 96, 042327 (2017).
  111. J. A. Vaccaro, F. Anselmi, H. M. Wiseman, and K. Jacobs, Tradeoff between extractable mechanical work, accessible entanglement, and ability to act as a reference system, under arbitrary superselection rules, Phys. Rev. A 77, 032114 (2008).
  112. F. Ares, S. Murciano, and P. Calabrese, Entanglement asymmetry as a probe of symmetry breaking, Nat. Commun. 14, 2036 (2023).
  113. L. Capizzi and M. Mazzoni, Entanglement asymmetry in the ordered phase of many-body systems: The Ising field theory, J. High Energy Phys. 12 (2023) 144.
  114. S. Yamashika, F. Ares, and P. Calabrese, Entanglement asymmetry and quantum Mpemba effect in two-dimensional free-fermion systems, Phys. Rev. B 110, 085126 (2024).
  115. M. Fossati, F. Ares, J. Dubail, and P. Calabrese, Entanglement asymmetry in CFT and its relation to non-topological defects, J. High Energy Phys. 05 (2024) 059.
  116. M. Chen and H.-H. Chen, Rényi entanglement asymmetry in (1+1)-dimensional conformal field theories, Phys. Rev. D 109, 065009 (2024).
  117. L. Capizzi and V. Vitale, A universal formula for the entanglement asymmetry of matrix product states, J. Phys. A 57, 45LT01 (2024).
  118. F. Ferro, F. Ares, and P. Calabrese, Non-equilibrium entanglement asymmetry for discrete groups: The example of the XY spin chain, J. Stat. Mech. (2024) 023101.
  119. F. Benini, V. Godet, and A. H. Singh, Entanglement asymmetry in conformal field theory and holography, Prog. Theor. Exp. Phys. 2025, 063B05 (2025).
  120. Y. Kusuki, S. Murciano, H. Ooguri, and S. Pal, Entanglement asymmetry and symmetry defects in boundary conformal field theory, J. High Energy Phys. 01 (2025) 057.
  121. Y. Guryanova, S. Popescu, A. J. Short, R. Silva, and P. Skrzypczyk, Thermodynamics of quantum systems with multiple conserved quantities, Nat. Commun. 7, 12049 (2016).
  122. M. Lostaglio, D. Jennings, and T. Rudolph, Thermodynamic resource theories, non-commutativity and maximum entropy principles, New J. Phys. 19, 043008 (2017).
  123. Y. Mitsuhashi, K. Kaneko, and T. Sagawa, Characterizing symmetry-protected thermal equilibrium by work extraction, Phys. Rev. X 12, 021013 (2022).
  124. R. Takagi and N. Shiraishi, Correlation in catalysts enables arbitrary manipulation of quantum coherence, Phys. Rev. Lett. 128, 240501 (2022).
  125. N. Shiraishi and R. Takagi, Arbitrary amplification of quantum coherence in asymptotic and catalytic transformation, Phys. Rev. Lett. 132, 180202 (2024).
  126. F. G. Brandão and G. Gour, Reversible framework for quantum resource theories, Phys. Rev. Lett. 115, 070503 (2015).
  127. F. G. S. L. Brandão and M. B. Plenio, A generalization of quantum Stein’s lemma, Commun. Math. Phys. 295, 791 (2010).
  128. M. Berta, F. G. S. L. Brandão, G. Gour, L. Lami, M. B. Plenio, B. Regula, and M. Tomamichel, On a gap in the proof of the generalised quantum Stein’s lemma and its consequences for the reversibility of quantum resources, Quantum 7, 1103 (2023).
  129. M. Hayashi and H. Yamasaki, The generalized quantum Stein’s lemma and the second law of quantum resource theories, Nat. Phys. 21, 1988 (2025).
  130. L. Lami, A solution of the generalised quantum Stein’s lemma, arXiv:2408.06410v1.
  131. B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics Vol. 222 (Springer International Publishing, Cham, 2015).
  132. A. W. Knapp, Lie Groups Beyond an Introduction (Springer Science & Business Media, New York, 2002).
  133. M. Hayashi, Quantum Information Theory: Mathematical Foundation, Graduate Texts in Physics (Springer, Berlin, Heidelberg, 2017).
  134. T. Ogawa and H. Nagaoka, Strong converse and Stein’s lemma in quantum hypothesis testing, IEEE Trans. Inf. Theory 46, 2428 (2000).
  135. R. D. Gill and S. Massar, State estimation for large ensembles, Phys. Rev. A 61, 042312 (2000).
  136. D. Pérez-García, M. M. Wolf, D. Petz, and M. B. Ruskai, Contractivity of positive and trace-preserving maps under Lp norms, J. Math. Phys. (N.Y.) 47, 083506 (2006).
  137. M. Guţă, J. Kahn, R. Kueng, and J. A. Tropp, Fast state tomography with optimal error bounds, J. Phys. A 53, 204001 (2020).
  138. M. Tomamichel, R. Colbeck, and R. Renner, Duality between smooth min- and max-entropies, IEEE Trans. Inf. Theory 56, 4674 (2010).
  139. M. Tomamichel, A framework for non-asymptotic quantum information theory, Doctoral Thesis, ETH Zurich, 2012.
  140. M. M. Wilde, Quantum Information Theory, 2nd ed. (Cambridge University Press, Cambridge, England, 2017).

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Phys. Rev. X 16, 031028– Published 5 August, 2026
  • Received 31 March 2025
  • Revised 23 March 2026
  • Accepted 5 June 2026
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