• Nielsen, 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).


    Google Scholar
     

  • Brandão, F. G. S. L. & Plenio, M. B. Entanglement theory and the second law of thermodynamics. Nat. Phys. 4, 873–877 (2008).

    Article 

    Google Scholar
     

  • Brandao, F. G. & Plenio, M. B. A reversible theory of entanglement and its relation to the second law. Commun. Math. Phys. 295, 829 (2010).

    Article 
    ADS 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Brandão, F. G. S. L. & Plenio, M. B. A generalization of quantum Stein’s lemma. Commun. Math. Phys. 295, 791 (2010).

    Article 
    ADS 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Brandão, F. G. S. L. & Gour, G. Reversible framework for quantum resource theories. Phys. Rev. Lett. 115, 070503 (2015).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Hiai, F. & Petz, D. The proper formula for relative entropy and its asymptotics in quantum probability. Commun. Math. Phys. 143, 99 (1991).

    Article 
    ADS 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Ogawa, T. & Nagaoka, H. Strong converse and stein’s lemma in quantum hypothesis testing. IEEE Trans. Inf. Theory 46, 2428 (2000).

    Article 
    ADS 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Fang, K., Gour, G. & Wang, X. Towards the ultimate limits of quantum channel discrimination and quantum communication. Sci. China Inf. Sci. 68, 180509 (2025).

    Article 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Berta, M. et al. 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).

    Article 

    Google Scholar
     

  • Berta, M. et al. The tangled state of quantum hypothesis testing. Nat. Phys. 20, 172 (2024).

    Article 
    ADS 

    Google Scholar
     

  • Yamasaki, H. & Kuroiwa, K. Generalized quantum Stein’s lemma: redeeming second law of resource theories. Preprint at https://arxiv.org/abs/2401.01926 (2024).

  • Hayashi, M. Optimal sequence of quantum measurements in the sense of stein’s lemma in quantum hypothesis testing. J. Phys. A Math. Gen. 35, 10759 (2002).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Nagaoka, H. & Hayashi, M. An information-spectrum approach to classical and quantum hypothesis testing for simple hypotheses. IEEE Trans. Inf. Theory 53, 534 (2007).

    Article 
    ADS 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Hayashi, M. Quantum Information Theory: Mathematical Foundation (Springer, 2016).

  • Lami, L. A solution of the generalised quantum Stein’s lemma. IEEE Trans. Inf. Theory 71, 4454–4454 (2025).

    Article 
    ADS 

    Google Scholar
     

  • Tomamichel, M. & Hayashi, M. Operational interpretation of Rényi information measures via composite hypothesis testing against product and Markov distributions. IEEE Trans. Inf. Theory 64, 1064 (2018).

    Article 
    ADS 
    MATH 

    Google Scholar
     

  • Nagaoka, H. in Asymptotic Theory of Quantum Statistical Inference (ed. Hayashi, M.) 64–65 (World Scientific, 2005).

  • Polyanskiy, Y., Poor, H. V. & Verdu, S. Channel coding rate in the finite blocklength regime. IEEE Trans. Inf. Theory 56, 2307 (2010).

    Article 
    ADS 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Polyanskiy, Y. Saddle point in the minimax converse for channel coding. IEEE Trans. Inf. Theory 59, 2576 (2013).

    Article 
    ADS 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Tyagi, H. & Watanabe, S. A bound for multiparty secret key agreement and implications for a problem of secure computing. In Advances in Cryptology—EUROCRYPT 2014 369–386 (Springer, 2014).

  • Tyagi, H. & Watanabe, S. Converses for secret key agreement and secure computing. IEEE Trans. Inf. Theory 61, 4809 (2015).

    Article 
    ADS 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Hayashi, M. & Owari, M. Tight asymptotic bounds on local hypothesis testing between a pure bipartite state and the white noise state. IEEE Trans. Inf. Theory 63, 4008 (2017).

    Article 
    ADS 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Renner, R. Security of quantum key distribution. Int. J. Quantum Inf. 06, 1 (2008).

    Article 
    MATH 

    Google Scholar
     

  • Renner, R. Symmetry of large physical systems implies independence of subsystems. Nat. Phys. 3, 645 (2007).

    Article 

    Google Scholar
     

  • Bluhm, A., Capel, A., Gondolf, P. & Pérez-Hernández, A. General continuity bounds for quantum relative entropies. In Proc. 2023 IEEE International Symposium on Information Theory (ISIT) 162–167 (IEEE, 2023).

  • Bluhm, A., Capel, A., Gondolf, P. & Pérez-Hernández, A. Continuity of quantum entropic quantities via almost convexity. IEEE Trans. Inf. Theory 69, 5869 (2023).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Zhao, Q., Liu, Y., Yuan, X., Chitambar, E. & Ma, X. One-shot coherence dilution. Phys. Rev. Lett. 120, 070403 (2018).

    Article 
    ADS 

    Google Scholar
     

  • Chitambar, E. Dephasing-covariant operations enable asymptotic reversibility of quantum resources. Phys. Rev. A 97, 050301 (2018).

    Article 
    ADS 

    Google Scholar
     

  • Horodecki, M., Horodecki, P. & Oppenheim, J. Reversible transformations from pure to mixed states and the unique measure of information. Phys. Rev. A 67, 062104 (2003).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Brandão, F. G. S. L., Horodecki, M., Oppenheim, J., Renes, J. M. & Spekkens, R. W. Resource theory of quantum states out of thermal equilibrium. Phys. Rev. Lett. 111, 250404 (2013).

    Article 
    ADS 

    Google Scholar
     

  • Brandão, F. G. S. L., Harrow, A. W., Lee, J. R. & Peres, Y. Adversarial hypothesis testing and a quantum Stein’s lemma for restricted measurements. IEEE Trans. Inf. Theory 66, 5037 (2020).

    Article 
    ADS 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Berta, M. & Majenz, C. Disentanglement cost of quantum states. Phys. Rev. Lett. 121, 190503 (2018).

    Article 
    ADS 

    Google Scholar
     

  • Gao, L. & Rahaman, M. Generalized Stein’s lemma and asymptotic equipartition property for subalgebra entropies. Preprint at https://arxiv.org/abs/2401.03090 (2024).

  • Hayashi, M. & Tomamichel, M. Correlation detection and an operational interpretation of the Rényi mutual information. J. Math. Phys. 57, 102201 (2016).

  • Regula, B. & Lami, L. Reversibility of quantum resources through probabilistic protocols. Nat. Commun. 15, 3096 (2024).

    Article 
    ADS 

    Google Scholar
     

  • Chen, Y.-A., Wang, X., Zhang, L. & Zhu, C. Reversible entanglement beyond quantum operations. Phys. Rev. Res. 7, 013297 (2025).

    Article 

    Google Scholar
     

  • Ganardi, R., Kondra, T. V., Ng, N. H. Y. & Streltsov, A. Second law of entanglement manipulation with entanglement battery. Phys. Rev. Lett. 135, 010202 (2025).

    Article 
    ADS 

    Google Scholar
     

  • Bennett, C. H., Bernstein, H. J., Popescu, S. & Schumacher, B. Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046 (1996).

    Article 
    ADS 

    Google Scholar
     

  • Hayden, P. M., Horodecki, M. & Terhal, B. M. The asymptotic entanglement cost of preparing a quantum state. J. Phys. A Math. Gen. 34, 6891 (2001).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Krueger, O. & Werner, R. F. Problem 20 in some open problems in quantum information theory. Preprint at https://arxiv.org/abs/quant-ph/0504166 (2005).

  • Kuroiwa, K. & Yamasaki, H. Asymptotically consistent measures of general quantum resources: discord, non-Markovianity, and non-Gaussianity. Phys. Rev. A 104, L020401 (2021).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Kuroiwa, K., Takagi, R., Adesso, G. & Yamasaki, H. Every quantum helps: operational advantage of quantum resources beyond convexity. Phys. Rev. Lett. 132, 150201 (2024).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Kuroiwa, K., Takagi, R., Adesso, G. & Yamasaki, H. Robustness- and weight-based resource measures without convexity restriction: multicopy witness and operational advantage in static and dynamical quantum resource theories. Phys. Rev. A 109, 042403 (2024).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Regula, B., Lami, L., Ferrari, G. & Takagi, R. Operational quantification of continuous-variable quantum resources. Phys. Rev. Lett. 126, 110403 (2021).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Lami, L., Regula, B., Takagi, R. & Ferrari, G. Framework for resource quantification in infinite-dimensional general probabilistic theories. Phys. Rev. A 103, 032424 (2021).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Ferrari, G., Lami, L., Theurer, T. & Plenio, M. B. Asymptotic state transformations of continuous variable resources. Commun. Math. Phys. 398, 291 (2023).

    Article 
    ADS 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Piani, M. Relative entropy of entanglement and restricted measurements. Phys. Rev. Lett. 103, 160504 (2009).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Brandao, F. G., Christandl, M. & Yard, J. Faithful squashed entanglement. Commun. Math. Phys. 306, 805 (2011).

    Article 
    ADS 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Boyd, S. & Vandenberghe, L. Convex Optimization (Cambridge Univ. Press, 2004).

  • Neumann, J. V. Zur theorie der gesellschaftsspiele. Math. Ann. 100, 295 (1928).

    Article 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Sion, M. On general minimax theorems. Pac. J. Math. 8, 171–176 (1958).

    Article 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Komiya, H. Elementary proof for Sion’s minimax theorem. Kodai Math. J. 11, 5 (1988).

    Article 
    ADS 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Takagi, R., Wang, K. & Hayashi, M. Application of the resource theory of channels to communication scenarios. Phys. Rev. Lett. 124, 120502 (2020).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Faist, P., Berta, M. & Brandão, F. Thermodynamic capacity of quantum processes. Phys. Rev. Lett. 122, 200601 (2019).

    Article 
    ADS 

    Google Scholar
     

  • Bennett, C., Shor, P., Smolin, J. & Thapliyal, A. Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem. IEEE Trans. Inf. Theory 48, 2637 (2002).

    Article 
    ADS 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Berta, M., Christandl, M. & Renner, R. The quantum reverse Shannon theorem based on one-shot information theory. Commun. Math. Phys. 306, 579 (2011).

    Article 
    ADS 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Bennett, C. H., Devetak, I., Harrow, A. W., Shor, P. W. & Winter, A. The quantum reverse Shannon theorem and resource tradeoffs for simulating quantum channels. IEEE Trans. Inf. Theory 60, 2926 (2014).

    Article 
    ADS 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Holevo, A. S. Quantum coding theorems. Russ. Math. Surv. 53, 1295 (1998).

    Article 
    MathSciNet 

    Google Scholar
     

  • Horodecki, M., Shor, P. W. & Ruskai, M. B. Entanglement breaking channels. Rev. Math. Phys. 15, 629 (2003).

    Article 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Chiribella, G., D’Ariano, G. M. & Perinotti, P. Quantum circuit architecture. Phys. Rev. Lett. 101, 060401 (2008).

    Article 
    ADS 

    Google Scholar
     

  • Chiribella, G., D’Ariano, G. M. & Perinotti, P. Theoretical framework for quantum networks. Phys. Rev. A 80, 022339 (2009).

    Article 
    ADS 
    MathSciNet 
    MATH 

    Google Scholar
     

  • Gour, G. & Winter, A. How to quantify a dynamical quantum resource. Phys. Rev. Lett. 123, 150401 (2019).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Gour, G., Marvian, I. & Spekkens, R. W. Measuring the quality of a quantum reference frame: the relative entropy of frameness. Phys. Rev. A 80, 012307 (2009).

    Article 
    ADS 

    Google Scholar
     

  • Hickey, A. & Gour, G. Quantifying the imaginarity of quantum mechanics. J. Phys. A Math. Theor. 51, 414009 (2018).

    Article 
    MathSciNet 

    Google Scholar
     

  • Fowler, A. G., Mariantoni, M., Martinis, J. M. & Cleland, A. N. Surface codes: towards practical large-scale quantum computation. Phys. Rev. A 86, 032324 (2012).

    Article 
    ADS 

    Google Scholar