-
Previous Article
On the non-Abelian group code capacity of memoryless channels
- AMC Home
- This Issue
-
Next Article
QC-LDPC construction free of small size elementary trapping sets based on multiplicative subgroups of a finite field
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 |
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.
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. |
[1] |
Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3351-3386. doi: 10.3934/dcdss.2020440 |
[2] |
Habibul Islam, Om Prakash, Ram Krishna Verma. New quantum codes from constacyclic codes over the ring $ R_{k,m} $. Advances in Mathematics of Communications, 2022, 16 (1) : 17-35. doi: 10.3934/amc.2020097 |
[3] |
Pablo Amster, Mariel Paula Kuna, Dionicio Santos. Stability, existence and non-existence of $ T $-periodic solutions of nonlinear delayed differential equations with $ \varphi $-Laplacian. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022070 |
[4] |
Chunyan Zhao, Chengkui Zhong, Xiangming Zhu. Existence of compact $ \varphi $-attracting sets and estimate of their attractive velocity for infinite-dimensional dynamical systems. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022051 |
[5] |
Magdalena Foryś-Krawiec, Jiří Kupka, Piotr Oprocha, Xueting Tian. On entropy of $ \Phi $-irregular and $ \Phi $-level sets in maps with the shadowing property. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1271-1296. doi: 10.3934/dcds.2020317 |
[6] |
Carlos García-Azpeitia. Relative periodic solutions of the $ n $-vortex problem on the sphere. Journal of Geometric Mechanics, 2019, 11 (3) : 427-438. doi: 10.3934/jgm.2019021 |
[7] |
Jianqin Zhou, Wanquan Liu, Xifeng Wang, Guanglu Zhou. On the $ k $-error linear complexity for $ p^n $-periodic binary sequences via hypercube theory. Mathematical Foundations of Computing, 2019, 2 (4) : 279-297. doi: 10.3934/mfc.2019018 |
[8] |
Cunsheng Ding, Chunming Tang. Infinite families of $ 3 $-designs from o-polynomials. Advances in Mathematics of Communications, 2021, 15 (4) : 557-573. doi: 10.3934/amc.2020082 |
[9] |
Florin Diacu, Shuqiang Zhu. Almost all 3-body relative equilibria on $ \mathbb S^2 $ and $ \mathbb H^2 $ are inclined. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1131-1143. doi: 10.3934/dcdss.2020067 |
[10] |
Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. Double well potential function and its optimization in the $N$ -dimensional real space-part Ⅱ. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1307-1328. doi: 10.3934/jimo.2016074 |
[11] |
Shu-Cherng Fang, David Y. Gao, Gang-Xuan Lin, Ruey-Lin Sheu, Wenxun Xing. Double well potential function and its optimization in the $N$ -dimensional real space-part Ⅰ. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1291-1305. doi: 10.3934/jimo.2016073 |
[12] |
David Hoff. Pointwise bounds for the Green's function for the Neumann-Laplace operator in $ \text{R}^3 $. Kinetic and Related Models, 2022, 15 (4) : 535-550. doi: 10.3934/krm.2021037 |
[13] |
María Anguiano, Alain Haraux. The $\varepsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors. Evolution Equations and Control Theory, 2017, 6 (3) : 345-356. doi: 10.3934/eect.2017018 |
[14] |
Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116 |
[15] |
Fanghui Ma, Jian Gao, Fang-Wei Fu. New non-binary quantum codes from constacyclic codes over $ \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $. Advances in Mathematics of Communications, 2019, 13 (3) : 421-434. doi: 10.3934/amc.2019027 |
[16] |
Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco. Upper bounds on the length function for covering codes with covering radius $ R $ and codimension $ tR+1 $. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2021074 |
[17] |
Yong Ren, Wensheng Yin. Quasi sure exponential stabilization of nonlinear systems via intermittent $ G $-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5871-5883. doi: 10.3934/dcdsb.2019110 |
[18] |
Yong Ren, Huijin Yang, Wensheng Yin. Weighted exponential stability of stochastic coupled systems on networks with delay driven by $ G $-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3379-3393. doi: 10.3934/dcdsb.2018325 |
[19] |
Lingyan Cheng, Ruinan Li, Liming Wu. Exponential convergence in the Wasserstein metric $ W_1 $ for one dimensional diffusions. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5131-5148. doi: 10.3934/dcds.2020222 |
[20] |
Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306 |
2020 Impact Factor: 0.935
Tools
Metrics
Other articles
by authors
[Back to Top]