A generalized quantum relative entropy

Luiza H. F. Andrade; Rui F. Vigelis; Charles C. Cavalcante

doi:10.3934/amc.2020063

Article Contents

2020, Volume 14, Issue 3: 413-422. Doi: 10.3934/amc.2020063

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A generalized quantum relative entropy

1.
Department of Natural Sciences, Mathematics and Statistics, Federal Rural University of the Semi-arid Region, Mossoró-RN, Brazil
2.
Computer Engineering, Campus Sobral, Federal University of Ceará, Sobral-CE, Brazil
3.
Department of Teleinformatics Engineering, Federal University of Ceará, Fortaleza-CE, Brazil

^* Corresponding author: Luiza H. F. Andrade
^* Corresponding author: Luiza H. F. Andrade

Received: November 30, 2018

Revised: August 31, 2019

Early access: January 2020

Published: August 2020

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Abstract

We propose a generalization of the quantum relative entropy by considering the geodesic on a manifold formed by all the invertible density matrices $ \mathcal{P} $. This geodesic is defined from a deformed exponential function $ \varphi $ which allows to work with a wider class of families of probability distributions. Such choice allows important flexibility in the statistical model. We show and discuss some properties of this proposed generalized quantum relative entropy.

Keywords:

Quantum relative entropy,
quantum information theory,
deformed exponential function,
$ \varphi $-families of probablity distributions.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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References

[1]	S. Abe, Nonadditive generalization of the quantum Kullback-Leibler divergence for measuring the degree of purification, Phys. Rev. A, 68 (2003), 032302. doi: 10.1103/PhysRevA.68.032302.
[2]	S.-i. Amari, A. Ohara and H. Matsuzoe, Geometry of deformed exponential families: Invariant, dually-flat and conformal geometries, Phys. A, 391 (2012), 4308-4319. doi: 10.1016/j.physa.2012.04.016.
[3]	L. Borland, A. R. Plastino and C. Tsallis, Information gain within nonextensive thermostatistics, Journal of Mathematical Physics, 39 (1998), 6490-6501. doi: 10.1063/1.532660.
[4]	A. Cena and G. Pistone, Exponential statistical manifold, Ann. Inst. Statist. Math., 59 (2007), 27-56. doi: 10.1007/s10463-006-0096-y.
[5]	D. C. de Souza, R. F. Vigelis and C. C. Cavalcante, Geometry induced by a generalization of rényi divergence, Entropy, 18 (2016), Paper No. 407, 16 pp. doi: 10.3390/e18110407.
[6]	S. Furuichi, On uniqueness theorems for Tsallis entropy and Tsallis relative entropy, IEEE Transactions on Information Theory, 51 (2005), 3638-3645. doi: 10.1109/TIT.2005.855606.
[7]	S. Furuichi, K. Yanagi and K. Kuriyama, Fundamental properties of Tsallis relative entropy, J. Math. Phys., 45 (2004), 4868-4877. doi: 10.1063/1.1805729.
[8]	M. R. Grasselli and R. F. Streater, On the uniqueness of the Chentsov metric in quantum information geometry, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 4 (2001), 173-182. doi: 10.1142/S0219025701000462.
[9]	K. Hoffman and R. Kunze, Linear Algebra, Second edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971.
[10]	G. Kaniadakis, Non-linear kinetics underlying generalized statistics, Physica A: Statistical Mechanics and its Applications, 296 (2001), 405-425. doi: 10.1016/S0378-4371(01)00184-4.
[11]	G. Kaniadakis, Statistical mechanics in the context of special relativity, Phys. Rev. E, 66 (2002), 056125, 17 pp. doi: 10.1103/PhysRevE.66.056125.
[12]	S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Statist., 22 (1951), 79-86. doi: 10.1214/aoms/1177729694.
[13]	J. Naudts, Estimators, escort probabilities, and $\phi$-exponential families in statistical physics, JIPAM. J. Inequal. Pure Appl. Math., 5 (2004), Art. 102, 15 pp.
[14]	M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000.
[15]	D. Petz, A survey of certain trace inequalities, Functional Analysis and Operator Theory, Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 30 (1994), 287-298.
[16]	D. Petz, Quantum Information Theory and Quantum Statistics, Theoretical and Mathematical Physics, Springer-Verlag, Berlin, 2008.
[17]	G. Pistone, $\kappa$-exponential models from the geometrical viewpoint, The European Physical Journal B, 70 (2009), 29-37.
[18]	G. Pistone and C. Sempi, An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one, Ann. Statist., 23 (1995), 1543-1561. doi: 10.1214/aos/1176324311.
[19]	C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Statist. Phys., 52 (1988), 479-487. doi: 10.1007/BF01016429.
[20]	C. Tsallis, What are the numbers that experiments provide, Quimica Nova, 17 (1994), 468-471.
[21]	S. Umarov, C. Tsallis and S. Steinberg, On a $q$-central limit theorem consistent with nonextensive statistical mechanics, Milan J. Math., 76 (2008), 307-328. doi: 10.1007/s00032-008-0087-y.
[22]	R. F. Vigelis and C. C. Cavalcante, On $\phi$-families of probability distributions, J. Theoret. Probab., 26 (2013), 870-884. doi: 10.1007/s10959-011-0400-5.