Advanced Search
Article Contents
Article Contents

A generalized quantum relative entropy

  • * Corresponding author: Luiza H. F. Andrade

    * Corresponding author: Luiza H. F. Andrade 
Abstract Full Text(HTML) Related Papers Cited by
  • 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.

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


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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. AmariA. 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. BorlandA. 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. FuruichiK. 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. ChuangQuantum 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. UmarovC. 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.
  • 加载中

Article Metrics

HTML views(485) PDF downloads(409) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint