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