Quantum f-divergences and error correction

Reviews in Mathematical Physics

Volumn 23, Issue 7, 2011, Pages 691-747

  Hiai, Fumio ^a   Mosonyi, Milán ^a,b   Petz, Dénes ^c   Bény, Cédric ^b  

^a TOHOKU UNIVERSITY   (Japan)

^b NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^c BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS   (Hungary)

Author keywords

Chernoff distance; f divergences; Hoeffding distances; operator convex functions; quasi entropy; R nyi relative entropies; Relative entropy; Schwarz maps; stochastic maps; substochastic maps

Indexed keywords

EID: 80052040382     PISSN: 0129055X     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0129055X11004412     Document Type: Article

References (53)

1. S. M. Ali and S. D. Silvey, A general class of coefficients of divergence of one distribution from another, J. Roy. Statist. Soc. Ser. B 28 (1966) 131-142.
2. H. Araki, On an inequality of Lieb and Thirring, Lett. Math. Phys. 19 (1990) 167-170.
3. K. M. R. Audenaert, J. Calsamiglia, Ll. Masanes, R. Munoz-Tapia, A. Acin, E. Bagan and F. Verstraete, Discriminating states: The quantum Chernoff bound, Phys. Rev. Lett. 98 (2007) 160501.
4. R. Bhatia, Matrix Analysis (Springer, 1997).
5. R. Bhatia, Positive Definite Matrices (Princeton University Press, 2007).
6. R. Blume-Kohout, H. K. Ng, D. Poulin and L. Viola, Information preserving structures: A general framework for quantum zero-error information, Phys. Rev. A 82 (2010) 062306.
7. T. Cover and J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 1991).
8. I. Csiszár, Information type measure of difference of probability distributions and indirect observations, Stud. Sci. Math. Hung. 2 (1967) 299-318.
9. I. Csiszár, Generalized cutoff rates and Rényi's information measures, IEEE Trans. Inform Theory 41 (1995) 26-34.
10. A. Datta, A condition for the nullity of quantum discord, arXiv:1003.5256.
11. E. G. Effros, A matrix convexity approach to some celebrated quantum inequalities, Proc. Natl. Acad. Sci. 106 (2009) 1006-1008.
12. I. Ekeland and R. Temam, Convex Analysis and Variational Problems (North- Holland, American Elsevier, 1976).
13. F. Hansen and G. K. Pedersen, Jensen's inequality for operators and Löwner's theorem, Math. Ann. 258 (1982) 229-241.
14. M. Hayashi, Quantum Information: An Introduction (Springer, 2006).
15. M. Hayashi, Error exponent in asymmetric quantum hypothesis testing and its application to classical-quantum channel coding, Phys. Rev. A 76 (2007) 062301.
16. M. Hayashi, private communication.
17. M. Hayashi, Symmetry and Quantum Information (Iwanami-Shoten, in press); in Japanese.
18. P. Hayden, R. Jozsa, D. Petz and A. Winter, Structure of states which satisfy strong subadditivity of quantum entropy with equality, Comm. Math. Phys. 246 (2004) 359-374. (Pubitemid 38545927)
19. F. Hiai, Matrix analysis: Matrix monotone functions, matrix means, and majorization (GSIS selected lectures), Interdiscip. Inform. Sci. 16 (2010) 139-248.
20. F. Hiai, unpublished.
21. F. Hiai and D. Petz, The proper formula for relative entropy and its asymptotics in quantum probability, Comm. Math. Phys. 143 (1991) 99-114.
22. F. Hiai, M. Mosonyi and T. Ogawa, Error exponents in hypothesis testing for correlated states on a spin chain, J. Math. Phys. 49 (2008) 032112.
23. A. Jenčová, Quantum hypothesis testing and sufficient subalgebras, Lett. Math. Phys. 93 (2010) 15-27.
24. A. Jenčová, Reversibility conditions for quantum operations, arXiv:1107.0453.
25. A. Jenčová and D. Petz, Sufficiency in quantum statistical inference, Comm. Math. Phys. 263 (2006) 259-276. (Pubitemid 43234482)
26. A. Jenčová and D. Petz, Sufficiency in quantum statistical inference. A survey with examples, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006) 331-351. (Pubitemid 44372565)
27. A. Jenčová and M. B. Ruskai, A unified treatment of convexity of relative entropy and related trace functions, with conditions for equality, Rev. Math. Phys. 22 (2010) 1099-1121.
28. A. Jenčová, D. Petz and J. Pitrik, Markov triplets on CCR algebras, Acta Sci. Math. (Szeged) 76 (2010) 27-50.
29. F. Kraus, Über konvexe matrixfunktionen, Math. Z. 41 (1936) 18-42.
30. A. Lesniewski and M. B. Ruskai, Monotone Riemannian metrics and relative entropy on non-commutative probability spaces, J. Math. Phys. 40 (1999) 5702-5724.
31. F. Liese and I. Vajda, Convex Statistical Distances (B. G. Teubner Verlagsgesellschaft, Leipzig, 1987).
32. F. Liese and I. Vajda, On divergences and informations in statistics and information theory, IEEE Trans. Inform. Theory 52 (2006) 4394-4412. (Pubitemid 46445281)
33. M. Mosonyi, Entropy, information and structure of composite quantum states, Ph.D. thesis, Catholic University of Leuven (2005), https://repository. cc.kuleuven.be/ dspace/handle/1979/41.
34. M. Mosonyi and D. Petz, Structure of sufficient quantum coarse grainings, Lett. Math. Phys. 68 (2004) 19-30. (Pubitemid 39583432)
35. M. Mosonyi and F. Hiai, On the quantum Renyi relative entropies and related capacity formulas, IEEE Trans. Inform. Theory 57 (2011) 2474-2487.
36. H. Nagaoka, The converse part of the theorem for quantum Hoeffding bound, preprint; quant-ph/0611289.
37. M. Nussbaum and A. Szkola, A lower bound of Chernoff type for symmetric quantum hypothesis testing, Ann. Statist. 37 (2009) 1040-1057.
38. T. Ogawa and H. Nagaoka, Strong converse and Stein's lemma in quantum hypothesis testing, IEEE Trans. Inform. Theory 47 (2000) 2428-2433. (Pubitemid 32090024)
39. T. Ogawa, Perfect quantum error-correcting condition revisited (2005); arXiv:quantph/ 0505167.
40. M. Ohya and D. Petz, Quantum Entropy and Its Use, 2nd edn. (Springer-Verlag, Heidelberg, 2004).
41. D. Petz, Quasi-entropies for states of a von Neumann algebra, Publ. RIMS. Kyoto Univ. 21 (1985) 781-800.
42. D. Petz, Quasi-entropies for finite quantum systems, Rep. Math. Phys. 23 (1986) 57-65.
43. D. Petz, Sufficiency of channels over von Neumann algebras Quart. J. Math. Oxford Ser. (2) 39(153) (1988) 97-108.
44. D. Petz, Monotonicity of quantum relative entropy revisited, Rev. Math. Phys. 15 (2003) 79-91. (Pubitemid 36344982)
45. D. Petz, Quantum Information Theory and Quantum Statistics (Springer, 2008).
46. D. Petz, From f-divergence to quantum quasi-entropies and their use, Entropy 12 (2010) 304-325.
47. A. Rényi, On measures of entropy and information, in Proc. 4th Berkeley Symp. on Math. Statist. Probability, Vol. 4, Berkeley, CA, USA (1961), pp. 547-561.
48. M. B. Ruskai, Inequalities for quantum entropy: A review with conditions for equality, J. Math. Phys. 43 (2002) 4358-4375.
49. N. Sharma, On the quantum f-relative entropy and generalized data processing inequalities (2009); arXiv:0906.4755.
50. M. Takesaki, Conditional expectations in von Neumann algebras, J. Funct. Anal. 9 (1972) 306-321.
51. M. Tomamichel, R. Colbeck and R. Renner, A fully quantum asymptotic equipartition property, IEEE Trans. Inform Theory 55 (2009) 5840-5847.
52. J. Tomiyama, On the geometry of positive maps in matrix algebras. II, Linear Algebra Appl. 69 (1985) 169-177.
53. A. Uhlmann, The "transition probability" in the state space of a *-algebra, Rep. Math. Phys. 9 (1976) 273-279.