Lower semicontinuity of relative entropy disturbance and its consequences

M. E. Shirokov

Sbornik. Mathematics, Tome 215 (2024) no. 11, pp. 1549-1581

Résumé

It is proved that the decrease of quantum relative entropy under the action of a quantum operation is a lower semicontinuous function of the pair of its arguments. This property implies, in particular, that the local discontinuity jumps of the quantum relative entropy do not increase under the action of quantum operations. It also implies the lower semicontinuity of the modulus of the joint convexity of quantum relative entropy (as a function of ensembles of quantum states). Various corollaries and applications of these results are considered. Bibliography: 42 titles.

Détail

Keywords: Hilbert space, trace-class operator, quantum state, lower semicontinuous function, quantum operation, strong convergence of quantum operations.

Bibliographie

[1] H. Umegaki, “Conditional expectation in an operator algebra. IV. Entropy and information”, Kōdai Math. Sem. Rep., 14:2 (1962), 59–85 | DOI | MR | Zbl

[2] B. Schumacher and M. D. Westmoreland, Relative entropy in quantum information theory, arXiv: quant-ph/0004045

[3] V. Vedral, “The role of relative entropy in quantum information theory”, Rev. Modern Phys., 74:1 (2002), 197–234 | DOI | MR | Zbl

[4] M. Ohya and D. Petz, Quantum entropy and its use, Theoret. Math. Phys., Corr. 2nd pr., Springer-Verlag, Berlin, 2004, 357 pp. | MR | Zbl

[5] A. Wehrl, “General properties of entropy”, Rev. Modern Phys., 50:2 (1978), 221–260 | DOI | MR

[6] A. S. Holevo, Statistical structure of quantum theory, Lect. Notes Phys. Monogr., 67, Springer-Verlag, Berlin, 2001, x+159 pp. | DOI | MR | Zbl

[7] G. Lindblad, “Completely positive maps and entropy inequalities”, Comm. Math. Phys., 40:2 (1975), 147–151 | DOI | MR | Zbl

[8] G. Lindblad, “Expectations and entropy inequalities for finite quantum systems”, Comm. Math. Phys., 39:2 (1974), 111–119 | DOI | MR | Zbl

[9] D. Petz, “Sufficiency of channels over von Neumann algebras”, Quart. J. Math. Oxford Ser. (2), 39:153 (1988), 97–108 | DOI | MR | Zbl

[10] D. Sutter, M. Tomamichel and A. W. Harrow, “Strengthened monotonicity of relative entropy via pinched Petz recovery map”, IEEE Trans. Inform. Theory, 62:5 (2016), 2907–2913 | DOI | MR | Zbl

[11] M. M. Wilde, “Recoverability in quantum information theory”, Proc. A, 471:2182 (2015), 20150338, 19 pp. | DOI | MR | Zbl

[12] M. Junge, R. Renner, D. Sutter, M. M. Wilde and A. Winter, “Universal recovery maps and approximate sufficiency of quantum relative entropy”, Ann. Henri Poincaré, 19:10 (2018), 2955–2978 | DOI | MR | Zbl

[13] M. E. Shirokov, “Strong convergence of quantum channels: continuity of the Stinespring dilation and discontinuity of the unitary dilation”, J. Math. Phys., 61:8 (2020), 082204, 14 pp. | DOI | MR | Zbl

[14] A. S. Holevo, Quantum systems, channels, information. A mathematical introduction, De Gruyter Stud. Math. Phys., 16, De Gruyter, Berlin, 2012, xiv+349 pp. | DOI | MR | Zbl

[15] M. E. Shirokov, “Convergence criterion for quantum relative entropy and its use”, Sb. Math., 213:12 (2022), 1740–1772 | DOI | MR | Zbl

[16] M. M. Wilde, Quantum information theory, Cambridge Univ. Press, Cambridge, 2013, xvi+655 pp. | DOI | MR | Zbl

[17] B. Simon, Operator theory, Compr. Course Anal., Part 4, Amer. Math. Soc., Providence, RI, 2015, xviii+749 pp. | DOI | MR | Zbl

[18] A. S. Holevo, “Some estimates for information quantity transmitted by quantum communication channel”, Problems Inform. Transmission, 9:3 (1973), 177–183 | MR | Zbl

[19] W. F. Stinespring, “Positive functions on $C^*$-algebras”, Proc. Amer. Math. Soc., 6:2 (1955), 211–216 | DOI | MR | Zbl

[20] G. F. Dell'Antonio, “On the limits of sequences of normal states”, Comm. Pure Appl. Math., 20:2 (1967), 413–429 | DOI | MR | Zbl

[21] M. J. Donald, “Further results on the relative entropy”, Math. Proc. Cambridge Philos. Soc., 101:2 (1987), 363–373 | DOI | MR | Zbl

[22] F. Buscemi and M. Horodecki, “Towards a unified approach to information-disturbance tradeoffs in quantum measurements”, Open Syst. Inf. Dyn., 16:1 (2009), 29–48 | DOI | MR | Zbl

[23] F. Buscemi, S. Das and M. M. Wilde, “Approximate reversibility in the context of entropy gain, information gain, and complete positivity”, Phys. Rev. A, 93:6 (2016), 062314, 11 pp. | DOI

[24] M. Berta, F. G. S. L. Brandao, C. Majenz and M. M. Wilde, Deconstruction and conditional erasure of quantum correlations, arXiv: 1609.06994

[25] Á. Capel, A. Lucia and D. Pérez-García, “Quantum conditional relative entropy and quasi-factorization of the relative entropy”, J. Phys. A, 51:48 (2018), 484001, 41 pp. | DOI | MR | Zbl

[26] T. S. Cubitt, M. B. Ruskai and G. Smith, “The structure of degradable quantum channels”, J. Math. Phys., 49:10 (2008), 102104, 27 pp. | DOI | MR | Zbl

[27] M. E. Shirokov, “Measures of correlations in infinite-dimensional quantum systems”, Sb. Math., 207:5 (2016), 724–768 | DOI | MR | Zbl

[28] M. E. Shirokov, “Correlation measures of a quantum state and information characteristics of a quantum channel”, J. Math. Phys., 64:11 (2023), 112201, 31 pp. | DOI | MR | Zbl

[29] M. E. Shirokov, “Convergence conditions for the quantum relative entropy and other applications of the generalized quantum Dini lemma”, Lobachevskii J. Math., 43:7 (2022), 1755–1777 | DOI | MR | Zbl

[30] M. Reed and B. Simon, Methods of modern mathematical physics, v. I, Functional analysis, Academic Press, Inc., New York–London, 1972, xvii+325 pp. | MR | Zbl

[31] M. E. Shirokov and A. V. Bulinski, “On quantum channels and operations preserving finiteness of the von Neumann entropy”, Lobachevskii J. Math., 41:12 (2020), 2383–2396 | DOI | MR | Zbl

[32] T. M. Cover and J. A. Thomas, Elements of information theory, 2nd ed., Wiley-Interscience [John Wiley Sons], Hoboken, NJ, 2006, xxiv+748 pp. | DOI | MR | Zbl

[33] C. M. Bishop, Pattern recognition and machine learning, Inf. Sci. Stat., Springer, New York, 2006, xx+738 pp. | MR | Zbl

[34] F. Nielsen, On the Kullback–Leibler divergence between location-scale densities, arXiv: 1904.10428

[35] V. I. Bogachev, Measure theory, v. II, Springer-Verlag, Berlin, 2007, xiv+575 pp. | DOI | MR | Zbl

[36] P. Billingsley, Convergence of probability measures, John Wiley Sons, Inc., New York–London–Sydney, 1968, xii+253 pp. | MR | Zbl

[37] Ch. Gerry and P. L. Knight, Introductory quantum optics, Cambridge Univ. Press, Cambridge, 2005, xiv+317 pp. | DOI

[38] R. J. Glauber, “Coherent and incoherent states of the radiation field”, Phys. Rev. (2), 131:6 (1963), 2766–2788 | DOI | MR | Zbl

[39] E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams”, Phys. Rev. Lett., 10:7 (1963), 277–279 | DOI | MR | Zbl

[40] M. E. Shirokov, “On lower semicontinuity of the quantum conditional mutual information and its corollaries”, Proc. Steklov Inst. Math., 313 (2021), 203–227 | DOI | MR | Zbl

[41] G. Lindblad, “Entropy, information and quantum measurements”, Comm. Math. Phys., 33:4 (1973), 305–322 | DOI | MR

[42] A. Bluhm, Á. Capel, P. Gondolf and A. Pérez-Hernández, “Continuity of quantum entropic quantities via almost convexity”, IEEE Trans. Inform. Theory, 69:9 (2023), 5869–5901 | DOI | MR | Zbl