Geodesic
Convergence criterion for quantum relative entropy and its use
Sbornik. Mathematics, Tome 213 (2022) no. 12, pp. 1740-1772 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A criterion and necessary conditions for the convergence (local continuity) of quantum relative entropy are obtained. Some applications of these results are considered. In particular, the preservation of the local continuity of quantum relative entropy under completely positive linear maps is established. Bibliography: 29 titles.
Keywords: separable Hilbert space, trace-class operator, quantum state, lower semicontinuous function, quantum operation, strong convergence of quantum operations. 
@article{SM_2022_213_12_a6,
     author = {M. E. Shirokov},
     title = {Convergence criterion for quantum relative entropy and its use},
     journal = {Sbornik. Mathematics},
     pages = {1740--1772},
     year = {2022},
     volume = {213},
     number = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2022_213_12_a6/}
}
TY  - JOUR
AU  - M. E. Shirokov
TI  - Convergence criterion for quantum relative entropy and its use
JO  - Sbornik. Mathematics
PY  - 2022
SP  - 1740
EP  - 1772
VL  - 213
IS  - 12
UR  - http://geodesic.mathdoc.fr/item/SM_2022_213_12_a6/
LA  - en
ID  - SM_2022_213_12_a6
ER  - 
%0 Journal Article
%A M. E. Shirokov
%T Convergence criterion for quantum relative entropy and its use
%J Sbornik. Mathematics
%D 2022
%P 1740-1772
%V 213
%N 12
%U http://geodesic.mathdoc.fr/item/SM_2022_213_12_a6/
%G en
%F SM_2022_213_12_a6
M. E. Shirokov. Convergence criterion for quantum relative entropy and its use. Sbornik. Mathematics, Tome 213 (2022) no. 12, pp. 1740-1772. http://geodesic.mathdoc.fr/item/SM_2022_213_12_a6/

[1] B. Schumacher and M. D. Westmoreland, “Relative entropy in quantum information theory”, Quantum computation and information (Washington, DC 2000), Contemp. Math., 305, Amer. Math. Soc., Providence, RI, 2002, 265–289 ; arXiv: quant-ph/0004045 | DOI | MR | Zbl

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

[3] A. Wehrl, “General properties of entropy”, Rev. Modern Phys., 50:2 (1978), 221–250 | DOI | MR

[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. S. Holevo, Quantum systems, channels, information, Moscow Center for Continuous Mathematical Education, Moscow, 2010, 328 pp.; English transl., 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

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

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

[8] J. Naudts, “Continuity of a class of entropies and relative entropies”, Rev. Math. Phys., 16:6 (2004), 809–822 | DOI | MR | Zbl

[9] M. E. Shirokov, “Convergence conditions for the quantum relative entropy and other applications of the deneralized quantum Dini lemma”, Lobachevskii J. Math., 43:7 (2022), 1755–1777 ; arXiv: 2205.09108 | DOI | Zbl

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

[11] L. Mirsky, “Symmetric gauge functions and unitarily invariant norms”, Quart. J. Math. Oxford Ser. (2), 11 (1960), 50–59 | DOI | MR | Zbl

[12] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information, Cambridge Univ. Press, Cambridge, 2000, xxvi+676 pp. | MR | Zbl

[13] W. Rudin, Principles of mathematical analysis, Internat. Ser. Pure Appl. Math., 3rd ed., McGraw-Hill Book Co., New York–Auckland–Düsseldorf, 1976, x+342 pp. | MR | Zbl

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

[15] O. Bratteli and D. W. Robinson, Operator algebras and quantum statistical mechanics, v. 1, Texts Monogr. Phys., $C^*$- and $W^*$-algebras, symmetry groups, decomposition of states, Springer-Verlag, New York–Heidelberg–Berlin, 1979, xii+500 pp. | DOI | MR | Zbl

[16] 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

[17] M. E. Shirokov and A. S. Holevo, “On approximation of infinite-dimensional quantum channels”, Probl. Peredachi Inf., 44:2 (2008), 3–22 ; English transl. in Problems Inform. Transmission, 44:2 (2008), 73–90 | MR | Zbl | DOI

[18] 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

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

[20] M. E. Shirokov, “Strong* convergence of quantum channels”, Quantum Inf. Process., 20:4 (2021), 145, 16 pp. | DOI | MR

[21] A. Winter, Energy-constrained diamond norm with applications to the uniform continuity of continuous variable channel capacities, arXiv: 1712.10267

[22] M. E. Shirokov, “Entropy characteristics of subsets of states. I”, Izv. Ross. Akad. Nauk Ser. Mat., 70:6 (2006), 193–222 ; English transl. in Izv. Math., 70:6 (2006), 1265–1292 | DOI | MR | Zbl | DOI

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

[24] M. E. Shirokov, Continuity of characteristics of composite quantum systems: a review, arXiv: 2201.11477

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

[26] 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

[27] N. I. Ahiezer and I. M. Glazman, Theory of linear operators in Hilbert space, 2nd ed., Nauka, Moscow, 1966, 543 pp. ; German transl., N. I. Achiezer und I. M. Glasmann, Theorie der linearen Operatoren im Hilbert-Raum, Math. Lehrbücher Monogr. I. Abt. Math. Lehrbücher, 4, Akademie-Verlag, Berlin, 1968, xvi+488 pp. | MR | Zbl | MR | Zbl

[28] M.-D. Choi, “Completely positive linear maps on complex matrices”, Linear Algebra Appl., 10:3 (1975), 285–290 | DOI | MR | Zbl

[29] A. Jamiolkowski, “Linear transformations which preserve trace and positive semidefiniteness of operators”, Rep. Math. Phys., 3:4 (1972), 275–278 | DOI | MR | Zbl