# An information-theoretic treatment of quantum dichotomies

1Graduate School of Informatics, Nagoya University, Nagoya, Japan
2Institute for Theoretical Physics, ETH Zurich, Switzerland
3Centre for Quantum Software and Information and School of Computer Science, University of Technology Sydney, Sydney

### Abstract

Given two pairs of quantum states, we want to decide if there exists a quantum channel that transforms one pair into the other. The theory of quantum statistical comparison and quantum relative majorization provides necessary and sufficient conditions for such a transformation to exist, but such conditions are typically difficult to check in practice. Here, by building upon work by Keiji Matsumoto, we relax the problem by allowing for small errors in one of the transformations. In this way, a simple sufficient condition can be formulated in terms of one-shot relative entropies of the two pairs. In the asymptotic setting where we consider sequences of state pairs, under some mild convergence conditions, this implies that the quantum relative entropy is the only relevant quantity deciding when a pairwise state transformation is possible. More precisely, if the relative entropy of the initial state pair is strictly larger compared to the relative entropy of the target state pair, then a transformation with exponentially vanishing error is possible. On the other hand, if the relative entropy of the target state is strictly larger, then any such transformation will have an error converging exponentially to one. As an immediate consequence, we show that the rate at which pairs of states can be transformed into each other is given by the ratio of their relative entropies. We discuss applications to the resource theories of athermality and coherence, where our results imply an exponential strong converse for general state interconversion.

### ► References

[1] P. Alberti and A. Uhlmann. A problem relating to positive linear maps on matrix algebras''. Reports on Mathematical Physics 18(2):163–176 (1980).
https:/​/​doi.org/​10.1016/​0034-4877(80)90083-x

[2] P. M. Alberti and A. Uhlmann. Stochasticity and partial order. volume 9 of Mathematics and Its Applications, Deutscher Verlag der Wissenschaften (1982).

[3] A. Anshu, M. Berta, R. Jain, and M. Tomamichel. A minimax approach to one-shot entropy inequalities''. Preprint, arXiv:1906.00333 (2019).
arXiv:1906.00333

[4] K. M. R. Audenaert, M. Mosonyi, and F. Verstraete. Quantum State Discrimination Bounds for Finite Sample Size''. Journal of Mathematical Physics 53(12):122205 (2012).
https:/​/​doi.org/​10.1063/​1.4768252

[5] D. Blackwell. Equivalent Comparisons of Experiments''. The Annals of Mathematical Statistics 24(2):265–272 (1953).
https:/​/​doi.org/​10.1214/​aoms/​1177729032

[6] F. G. S. L. Brandao, M. Horodecki, N. H. Y. Ng, J. Oppenheim, and S. Wehner. The Second Laws of Quantum Thermodynamics''. Proceedings of the National Academy of Sciences USA 112(11):3275–3279 (2014).
https:/​/​doi.org/​10.1073/​pnas.1411728112

[7] F. Buscemi. Fully quantum second-law–like statements from the theory of statistical comparisons''. Preprint, arXiv:1505.00535v1.
arXiv:1505.00535v1

[8] F. Buscemi. Comparison of Quantum Statistical Models: Equivalent Conditions for Sufficiency''. Communications in Mathematical Physics 310(3):625–647 (2012).
https:/​/​doi.org/​10.1007/​s00220-012-1421-3

[9] F. Buscemi and N. Datta. The Quantum Capacity of Channels With Arbitrarily Correlated Noise''. IEEE Transactions on Information Theory 56(3):1447–1460 (2010).
https:/​/​doi.org/​10.1109/​TIT.2009.2039166

[10] F. Buscemi and G. Gour. Quantum relative Lorenz curves''. Physical Review A 95(1) (2017).
https:/​/​doi.org/​10.1103/​physreva.95.012110

[11] A. Chefles. Deterministic Quantum State Transformations''. Physics Letters A 270(1-2):14–19 (2000).
https:/​/​doi.org/​10.1016/​S0375-9601(00)00291-7

[12] A. Chefles, R. Jozsa, and A. Winter. On the Existence of Physical Transformations between Sets of Quantum States''. International Journal of Quantum Information 02(01):11–21 (2004).
https:/​/​doi.org/​10.1142/​S0219749904000031

[13] H.-C. Cheng and M.-H. Hsieh. Moderate deviation analysis for classical-quantum channels and quantum hypothesis testing''. IEEE Transactions on Information Theory 64(2):1385–1403 (2018).
https:/​/​doi.org/​10.1109/​TIT.2017.2781254

[14] E. Chitambar. Dephasing-covariant operations enable asymptotic reversibility of quantum resources''. Physical Review A 97(5) (2018).
https:/​/​doi.org/​10.1103/​physreva.97.050301

[15] E. Chitambar and G. Gour. Critical Examination of Incoherent Operations and a Physically Consistent Resource Theory of Quantum Coherence''. Physical Review Letters 117(3) (2016).
https:/​/​doi.org/​10.1103/​physrevlett.117.030401

[16] E. Chitambar and G. Gour. Quantum resource theories''. Reviews of Modern Physics 91(2) (2019).
https:/​/​doi.org/​10.1103/​revmodphys.91.025001

[17] C. T. Chubb, V. Y. F. Tan, and M. Tomamichel. Moderate Deviation Analysis for Classical Communication over Quantum Channels''. Communications in Mathematical Physics 355(3):1283–1315 (2017).
https:/​/​doi.org/​10.1007/​s00220-017-2971-1

[18] C. T. Chubb, M. Tomamichel, and K. Korzekwa. Beyond the thermodynamic limit: finite-size corrections to state interconversion rates''. Quantum 2:108 (2018).
https:/​/​doi.org/​10.22331/​q-2018-11-27-108

[19] C. T. Chubb, M. Tomamichel, and K. Korzekwa. Moderate deviation analysis of majorization-based resource interconversion''. Physical Review A 99(3):032332 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.032332

[20] J. E. Cohen, J. H. B. Kempermann, and G. Zbaganu. Comparisons of Stochastic Matrices with Applications in Information Theory, Statistics, Economics and Population. Birkhäuser (1998).

[21] G. Dahl. Matrix majorization''. Linear Algebra and its Applications 288:53 – 73 (1999).
https:/​/​doi.org/​10.1016/​S0024-3795(98)10175-1

[22] N. Datta. Min- and Max- Relative Entropies and a New Entanglement Monotone''. IEEE Transactions on Information Theory 55(6):2816–2826 (2009).
https:/​/​doi.org/​10.1109/​TIT.2009.2018325

[23] G. Gour, D. Jennings, F. Buscemi, R. Duan, and I. Marvian. Quantum majorization and a complete set of entropic conditions for quantum thermodynamics''. Nature Communications 9(1) (2018).
https:/​/​doi.org/​10.1038/​s41467-018-06261-7

[24] G. H. Hardy, J. E. Littlewood, and G. Pólya. Inequalities. Cambridge University Press (1934).

[25] F. Hiai and D. Petz. The Proper Formula for Relative Entropy and its Asymptotics in Quantum Probability''. Communications in Mathematical Physics 143(1):99–114 (1991).
https:/​/​doi.org/​10.1007/​BF02100287

[26] M. Horodecki and J. Oppenheim. Fundamental limitations for quantum and nanoscale thermodynamics''. Nature Communications 4(1) (2013).
https:/​/​doi.org/​10.1038/​ncomms3059

[27] A. Jenčová. Comparison of Quantum Binary Experiments''. Reports on Mathematical Physics 70(2):237–249 (2012).
https:/​/​doi.org/​10.1016/​s0034-4877(12)60043-3

[28] A. Jenčová. Comparison of quantum channels and statistical experiments''. In Proc. IEEE ISIT 2016, pages 2249–2253, (2016).
https:/​/​doi.org/​10.1109/​ISIT.2016.7541699

[29] J. Körner and K. Marton. Comparison of two noisy channels''. Colloquia Mathematica Societatis Janos Bolyai, Topics in Information Theory 16:411–424, (1977).

[30] K. Korzekwa, C. T. Chubb, and M. Tomamichel. Avoiding Irreversibility: Engineering Resonant Conversions of Quantum Resources''. Physical Review Letters 122(11):110403 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.110403

[31] W. Kumagai and M. Hayashi. Second Order Asymptotics of Optimal Approximate Conversion for Probability Distributions and Entangled States and Its Application to LOCC Cloning''. Preprint, arXiv:1306.4166 (2013).
arXiv:1306.4166

[32] L. Le Cam. Sufficiency and Approximate Sufficiency''. The Annals of Mathematical Statistics 35(4):1419–1455 (1964).
https:/​/​doi.org/​10.1214/​aoms/​1177700372

[33] K. Li. Second-Order Asymptotics for Quantum Hypothesis Testing''. Annals of Statistics 42(1):171–189 (2014).
https:/​/​doi.org/​10.1214/​13-AOS1185

[34] A. W. Marshall, I. Olkin, and B. C. Arnold. Inequalities: Theory of Majorization and Its Applications. Springer (2011).
https:/​/​doi.org/​10.1007/​978-0-387-68276-1

[35] I. Marvian and R. W. Spekkens. How to quantify coherence: Distinguishing speakable and unspeakable notions''. Physical Review A 94(5) (2016).
https:/​/​doi.org/​10.1103/​physreva.94.052324

[36] K. Matsumoto. A quantum version of randomization criterion''. Preprint, arXiv:1012.2650 (2010).
arXiv:1012.2650

[37] K. Matsumoto. Reverse Test and Characterization of Quantum Relative Entropy''. Preprint, arXiv:1010.1030 (2010).
arXiv:1010.1030

[38] K. Matsumoto. An example of a quantum statistical model which cannot be mapped to a less informative one by any trace preserving positive map''. Preprint, arXiv:1409.5658 (2014).
arXiv:1409.5658

[39] K. Matsumoto. On the condition of conversion of classical probability distribution families into quantum families''. Preprint, arXiv:1412.3680 (2014).
arXiv:1412.3680

[40] M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel. On Quantum Rényi Entropies: A New Generalization and Some Properties''. Journal of Mathematical Physics 54(12):122203 (2013).
https:/​/​doi.org/​10.1063/​1.4838856

[41] M. A. Nielsen. Conditions for a Class of Entanglement Transformations''. Physical Review Letters 83(2):436–439 (1999).
https:/​/​doi.org/​10.1103/​physrevlett.83.436

[42] T. Ogawa and H. Nagaoka. Strong Converse and Stein's Lemma in Quantum Hypothesis Testing''. IEEE Transactions on Information Theory 46(7):2428–2433 (2000).
https:/​/​doi.org/​10.1109/​18.887855

[43] D. Petz. Quasi-entropies for Finite Quantum Systems''. Reports on Mathematical Physics 23(1):57–65 (1986).
https:/​/​doi.org/​10.1016/​0034-4877(86)90067-4

[44] D. Reeb, M. J. Kastoryano, and M. M. Wolf. Hilbert's projective metric in quantum information theory''. Journal of Mathematical Physics 52(8):082201 (2011).
https:/​/​doi.org/​10.1063/​1.3615729

[45] B. Regula, K. Fang, X. Wang, and G. Adesso. One-Shot Coherence Distillation''. Physical Review Letters 121(1) (2018).
https:/​/​doi.org/​10.1103/​physrevlett.121.010401

[46] B. Regula, V. Narasimhachar, F. Buscemi, and M. Gu. Coherence manipulation with dephasing-covariant operations''. Preprint, arXiv:1907.08606 (2019).
arXiv:1907.08606

[47] J. M. Renes. Relative submajorization and its use in quantum resource theories''. Journal of Mathematical Physics 57(12):122202 (2016).
https:/​/​doi.org/​10.1063/​1.4972295

[48] R. Renner. Security of Quantum Key Distribution. PhD thesis, ETH Zurich, (2005). Available at arXiv:quant-ph/​0512258.
arXiv:quant-ph/0512258

[49] C. E. Shannon. A note on a partial ordering for communication channels''. Information and control 1(4):390–397 (1958).
https:/​/​doi.org/​10.1016/​S0019-9958(58)90239-0

[50] V. Siddhu and R. B. Griffiths. Degradable Quantum Channels using Pure-State to Product-of-Pure-State Isometries''. Physical Review A 94(5):052331 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.052331

[51] M. Tomamichel. Quantum Information Processing with Finite Resources — Mathematical Foundations. volume 5 of SpringerBriefs in Mathematical Physics, Springer International Publishing (2016).
https:/​/​doi.org/​10.1007/​978-3-319-21891-5

[52] M. Tomamichel, R. Colbeck, and R. Renner. A Fully Quantum Asymptotic Equipartition Property''. IEEE Transactions on Information Theory 55(12):5840–5847 (2009).
https:/​/​doi.org/​10.1109/​TIT.2009.2032797

[53] M. Tomamichel and M. Hayashi. A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks''. IEEE Transactions on Information Theory 59(11):7693–7710 (2013).
https:/​/​doi.org/​10.1109/​TIT.2013.2276628

[54] E. Torgersen. Comparison of statistical experiments. volume 36 of Encyclopedia of Mathematics and its Applications, Cambridge University Press (1991).
https:/​/​doi.org/​10.1017/​CBO9780511666353

[55] E. N. Torgersen. Comparison of experiments when the parameter space is finite''. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 16(3):219–249 (1970).
https:/​/​doi.org/​10.1007/​BF00534598

[56] X. Wang and M. M. Wilde. Resource theory of asymmetric distinguishability''. Preprint, arXiv:1905.11629 (2019).
arXiv:1905.11629

[57] M. M. Wilde, A. Winter, and D. Yang. Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy''. Communications in Mathematical Physics 331(2):593–622 (2014).
https:/​/​doi.org/​10.1007/​s00220-014-2122-x

### Cited by

[1] Bartosz Regula, Kaifeng Bu, Ryuji Takagi, and Zi-Wen Liu, "Benchmarking one-shot distillation in general quantum resource theories", Physical Review A 101 6, 062315 (2020).

[2] Bartosz Regula, Varun Narasimhachar, Francesco Buscemi, and Mile Gu, "Coherence manipulation with dephasing-covariant operations", Physical Review Research 2 1, 013109 (2020).

[3] Xin Wang and Mark M. Wilde, "Resource theory of asymmetric distinguishability", Physical Review Research 1 3, 033170 (2019).

[4] Philippe Faist, Takahiro Sagawa, Kohtaro Kato, Hiroshi Nagaoka, and Fernando G. S. L. Brandão, "Macroscopic Thermodynamic Reversibility in Quantum Many-Body Systems", Physical Review Letters 123 25, 250601 (2019).

[5] Soorya Rethinasamy and Mark M. Wilde, "Relative Entropy and Catalytic Relative Majorization", arXiv:1912.04254.

[6] Wenbin Zhou and Francesco Buscemi, "General state transitions with exact resource morphisms: a unified resource-theoretic approach", arXiv:2005.09188.

[7] Michele Dall'Arno, Francesco Buscemi, and Valerio Scarani, "Extension of the Alberti-Ulhmann criterion beyond qubit dichotomies", arXiv:1910.04294.

[8] Christopher Perry, Péter Vrana, and Albert H. Werner, "The semiring of dichotomies and asymptotic relative submajorization", arXiv:2004.10587.

The above citations are from Crossref's cited-by service (last updated successfully 2020-07-11 08:25:33) and SAO/NASA ADS (last updated successfully 2020-07-11 08:25:34). The list may be incomplete as not all publishers provide suitable and complete citation data.