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August  2020, 14(3): 413-422. doi: 10.3934/amc.2020063

A generalized quantum relative entropy

Luiza H. F. Andrade 1,, , Rui F. Vigelis 2,3, and  Charles C. Cavalcante 2,3,

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

Received  December 2018 Revised  September 2019 Published  August 2020 Early access  January 2020

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.
Citation: Luiza H. F. Andrade, Rui F. Vigelis, Charles C. Cavalcante. A generalized quantum relative entropy. Advances in Mathematics of Communications, 2020, 14 (3) : 413-422. doi: 10.3934/amc.2020063
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.

show all references

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.

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