Bounds of the Pinsker and Fannes Types on the Tsallis Relative Entropy

Mathematical Physics Analysis and Geometry

Volumn 16, Issue 3, 2013, Pages 213-228

  Rastegin, Alexey E ^a  

^a IRKUTSK STATE UNIVERSITY   (Russian Federation)

Author keywords

convexity; Fannes inequality; Fano inequality; Pinsker inequality; R nyi entropy; trace distance; Tsallis entropy

Indexed keywords

EID: 84881610926     PISSN: 13850172     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11040-013-9128-z     Document Type: Article

References (48)

1. Audenaert, K.: Subadditivity of q-entropies for q > 1. J. Math. Phys. 48, 083507 (2007).
2. Audenaert, K. M. R., Eisert, J.: Continuity bounds on the quantum relative entropy. J. Math. Phys. 46, 102104 (2005).
3. Audenaert, K. M. R., Eisert, J.: Continuity bounds on the quantum relative entropy - II. J. Math. Phys. 52, 112201 (2011).
4. Bhatia, R.: Positive definite matrices. Princeton University Press, Princeton (2007).
5. Borland, L., Plastino, A. R., Tsallis, C.: Information gain within nonextensive thermostatistics. J. Math. Phys. 39, 6490-6501 (1998). 40, 2196(E) (1999).
6. Bratteli, O., Robinson, D. W.: Operator algebras and quantum statistical mechanics 2. Equilibrium states. Models in quantum statistical mechanics. Springer, Berlin (2002).
7. Busch, P.: Stochastic isometries in quantum mechanics. Math. Phys. Anal. Geom. 2, 83-106 (1999).
8. Cai, L., Hansen, F.: Metric-adjusted skew information: convexity and restricted forms of superadditivity. Lett Math. Phys. 93, 1-13 (2010).
9. Csiszár, I.: A note on Jensen's inequality. Studia Sci. Math. Hung. 1, 227-230 (1966).
10. Csiszár, I.: Information-type measures of difference of probability distributions and indirect observations. Studia Sci Math. Hung. 2, 299-318 (1967).
11. Csiszár, I.: Axiomatic characterizations of information measures. Entropy 10, 261-273 (2008).
12. Fannes, M.: A continuity property of entropy density for spin lattice systems. Commun Math. Phys. 31, 291-294 (1973).
13. Fano, R. M.: Transmission of information: a statistical theory of communications. MIT Press and John Wiley & Sons, New York (1961).
14. Fedotov, A., Harremoës, P., Topsøe, F.: Refinements of Pinsker inequality. IEEE Trans. Inf. Theory 49, 1491-1498 (2003).
15. Fuchs, C. A., van de Graaf, J.: Cryptographic distinguishability measures for quantum mechanical states. IEEE Trans. Inf. Theory 45, 1216-1227 (1999).
16. Furuichi, S.: Information theoretical properties of Tsallis entropies. J. Math. Phys. 47, 023302 (2006).
17. Furuichi, S., Yanagi, K., Kuriyama, K.: Fundamental properties of Tsallis relative entropy. J. Math. Phys. 45, 4868-4877 (2004).
18. Furuichi, S., Yanagi, K., Kuriyama, K.: A generalized Fannes' inequality. J. Inequal. Pure Appl. Math. 8(1), 5 (2007).
19. Gell-Mann, M., Tsallis, C. (eds.): Nonextensive Entropy - Interdisciplinary Applications. Oxford University Press, Oxford (2004).
20. Gilardoni, G.: On Pinsker's and Vajda's type inequalities for Csiszár's f -divergence. IEEE Trans. Inf. Theory 56, 5377-5386 (2010).
21. Hansen, F.: The Wigner-Yanase entropy is not subadditive. J. Stat. Phys. 126, 643-648 (2007).
22. Hiai, F., Mosonyi, M., Petz, D., Bény, C.: Quantum f-divergences and error correction. Rev. Math. Phys. 23, 691-747 (2011).
23. Hiai, F., Ohya, M., Tsukada, M.: Sufficiency, KMS condition and relative entropy in von Neumann algebras. Pacific J. Math. 96, 99-109 (1981).
24. Jenčová, A., Ruskai, M. B.: A unified treatment of convexity of relative entropy and related trace functions, with conditions for equality. Rev. Math. Phys. 22, 1099-1121 (2010).
25. Kullback, S., Leibler, R. A.: On information and sufficiency. Ann. Math. Stat. 22, 79-86 (1951).
26. Lindvall, T.: Lectures on the Coupling Method. John Wiley & Sons, New York (1992).
27. Majerník, V., Majerníková, E.: The determination of bounds of the β-entropic sum of two noncommuting observables. Rep. Math. Phys. 47, 381-392 (2001).
28. Nielsen, M. A., Chuang, I. L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000).
29. Petz, D.: Quasi-entropies for finite quantum systems. Rep. Math. Phys. 21, 57-65 (1986).
30. Petz, D.: Quantum Information Theory and Quantum Statistics. Springer, Berlin (2008).
31. Petz, D.: From f -divergence to quantum quasi-entropies and their use. Entropy 12, 304-325 (2010).
32. Raggio, G. A.: Properties of q-entropies. J. Math. Phys. 36, 4785-4791 (1995).
33. Rastegin, A. E.: Continuity and stability of partial entropic sums. Lett. Math. Phys. 94, 229-242 (2010).
34. Rastegin, A. E.: Bounds on Shannon distinguishability in terms of partitioned measures. Quantum Inf. Process. 10, 123-138 (2011).
35. Rastegin, A. E.: Entropic uncertainty relations for extremal unravelings of super-operators. J. Phys. A: Math. Theor. 44, 095303 (2011).
36. Rastegin, A. E.: Some general properties of unified entropies. J. Stat. Phys. 143, 1120-1135 (2011).
37. Rastegin, A. E.: Upper continuity bounds on the relative q-entropy for q > 1. J. Math. Phys. 52, 062203 (2011).
38. Rastegin, A. E.: Notes on entropic uncertainty relations beyond the scope of Riesz's theorem. Int. J. Theor. Phys. 51, 1300-1315 (2012).
39. Rastegin, A. E.: Fano type quantum inequalities in terms of q-entropies. Quantum Inf. Process. 11, 1895-1910 (2012).
40. Rockafellar, R. T.: Convex Analysis. Princeton University Press, Princeton (1970).
41. Ruskai, M. B.: Inequalities for quantum entropy: A review with conditions for equality. J. Math. Phys. 43, 4358-4375 (2002).
42. Ruskai, M. B., Stillinger, F. M.: Convexity inequalities for estimating free energy and relative entropy. J. Phys. A: Math. Gen. 23, 2421-2437 (1990).
43. Seiringer, R.: On the failure of subadditivity of the Wigner-Yanase entropy. Lett. Math. Phys. 80, 285-288 (2007).
44. Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 32, 479-487 (1988).
45. Uhlmann, A.: Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory. Commun. Math. Phys. 54, 21-32 (1977).
46. Vedral, V.: The role of relative entropy in quantum information theory. Rev. Mod. Phys. 74, 197-234 (2002).
47. Zhang, Z.: Uniform estimates on the Tsallis entropies. Lett. Math. Phys. 80, 171-181 (2007).
48. Zozor, S., Portesi, M., Vignat, C.: Some extensions of the uncertainty principle. Physica A 387, 4800-4808 (2008).