关于量子熵的若干研究
摘要
熵理论在科学史上是一个极其重要的里程碑.从它产生至今已经一百五十余年了,但我们仍然难以对其进行一个科学的描述.伴随着热力学和统计力学的发展,熵的研究显得越发重要,且它的物理意义变得越发隐晦.
本文首先研究了序列效应代数的熵理论,主要结果如下：
●利用序列效应代数理论,本文提出了量子逻辑的分割以及分割的加细的概念,并给出了各种熵的定义.
●本文研究了序列效应代数的各种熵的关系及性质.
●本文定义和研究了态关于分割的相对熵.
●通过熵的韦恩图,我们说明了序列熵是经典熵的推广.特别地,通过量子测量的实例,本文说明了序列效应代数的熵及态关于分割的相对熵极其依赖于它的序列积的顺序.
其次,我们引进并研究了经典和量子的统一(r,s)-相对熵,这种相对熵是若干重要的广义相对熵的推广.特别地,本文的性质是建立在可分的量子系统下的,而不仅仅适用于有限维.其中,尤其重要的性质是,可分量子系统的量子统一(r,s)-相对熵在保迹完全正定映射下是不增的.
The entropy theory in the history of science is an extremely important mile-stone. More than one hundred and fifty years have passed since it generates, but we are still difficult to give them a scientific description. Accompanied by the develop-ment of thermodynamics and statistical mechanics, the study of the entropy theory becomes increasingly important, and its physical meaning has become increasingly obscure.
First, in this paper, we studies the entropy theory of the sequential effect algebra. Main results are as follows:
●Making use of the sequential effect algebra theory, we propose the definitions of the partition and the refinement of partitions, and present different kinds of entropies.
●We investigate the relations and properties of different entropies of the se-quential effect algebra.
●We define and study the relative entropy of states about partition,
●Using the venn diagram, we show that the entropies of sequential effect al-gebra are the generalization of the classical entropies. In particular, by some properties and examples, we display that the entropies of the partitions of sequential effect algebra and relative entropies of states about partition ex-tremely dependent on the order of its sequence product.
Second, we also introduce and study the classical and quantum unified (r, s)-relative entropies, which are the generalization of some important generalized rel-ative entropies. Particularly, our properties are based on the separable quantum system, not only suit for the finite dimensional quantum system. The most im-portant one in those properties is that quantum unified (r, s)-relative entropy for a separable quantum system is decreasing under a trace-preserving completely posi-tive mapping.
本文首先研究了序列效应代数的熵理论,主要结果如下：
●利用序列效应代数理论,本文提出了量子逻辑的分割以及分割的加细的概念,并给出了各种熵的定义.
●本文研究了序列效应代数的各种熵的关系及性质.
●本文定义和研究了态关于分割的相对熵.
●通过熵的韦恩图,我们说明了序列熵是经典熵的推广.特别地,通过量子测量的实例,本文说明了序列效应代数的熵及态关于分割的相对熵极其依赖于它的序列积的顺序.
其次,我们引进并研究了经典和量子的统一(r,s)-相对熵,这种相对熵是若干重要的广义相对熵的推广.特别地,本文的性质是建立在可分的量子系统下的,而不仅仅适用于有限维.其中,尤其重要的性质是,可分量子系统的量子统一(r,s)-相对熵在保迹完全正定映射下是不增的.
The entropy theory in the history of science is an extremely important mile-stone. More than one hundred and fifty years have passed since it generates, but we are still difficult to give them a scientific description. Accompanied by the develop-ment of thermodynamics and statistical mechanics, the study of the entropy theory becomes increasingly important, and its physical meaning has become increasingly obscure.
First, in this paper, we studies the entropy theory of the sequential effect algebra. Main results are as follows:
●Making use of the sequential effect algebra theory, we propose the definitions of the partition and the refinement of partitions, and present different kinds of entropies.
●We investigate the relations and properties of different entropies of the se-quential effect algebra.
●We define and study the relative entropy of states about partition,
●Using the venn diagram, we show that the entropies of sequential effect al-gebra are the generalization of the classical entropies. In particular, by some properties and examples, we display that the entropies of the partitions of sequential effect algebra and relative entropies of states about partition ex-tremely dependent on the order of its sequence product.
Second, we also introduce and study the classical and quantum unified (r, s)-relative entropies, which are the generalization of some important generalized rel-ative entropies. Particularly, our properties are based on the separable quantum system, not only suit for the finite dimensional quantum system. The most im-portant one in those properties is that quantum unified (r, s)-relative entropy for a separable quantum system is decreasing under a trace-preserving completely posi-tive mapping.
引文
[1]冯端,冯少彤.溯源探幽：熵的世界.科学出版社,书号：9787030155481,北京,2005.
[2]汤苏野.熵：一个世纪之谜的解析(第2版).中国科学技术大学出版社,书号：9787312021268,合肥,2008.
[3]M. Ohya and D. Petz, Quantum Entropy and its Use, Springer-Verlag, Berlin, 1991.
[4]H. J. Yuan, Entropy of Partitions on Quantum Logic, Commun. Theor. Phys. 43, (2005),437-439.
[5]J. L. Kelly, General Topology, Springer-Verlag, New York,1955.
[6]G. Birkhoff and J. von Neumann, The logic of quantum mechanics, Ann. Math. 37, (1936),823-834.
[7]A. Dvurecenskij and S. Pulmannova, New Trends in Quantum Structures, Kluwer,2002.
[8]李丞祖.量子通信和量子计算.国防科技大学出版社,书号：604577,长沙,2001.
[9]S. Kullback and R. A. Leibler. On information and sufficiency, Ann. Math. Stat.22, (1951),79-86.
[10]T. M. Cover and J. A. Thomas, Elements of Information Theory, Wiley, New York,1991
[11]钱伯出.量子力学.电子工业出版社,书号：7-5053-1934-5,北京,1993.
[12]曾谨严.量子力学(卷Ⅰ).第四版,科学出版社,书号：9787030181398,北京,2007.
[13]曾谨严.量子力学(卷Ⅱ),第四版,科学出版社,书号：9787030190215,北京,2007.
[14]周世勋.量子力学教程.高等教育出版社,书号：7040013223,北京,2008.
[15]P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford University Press, London,1958.
[16]L. D. Landau, Non-relativistic Theory, Pergamon, London,1977.
[17]M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Infor-mation, Cambrige University Press, Cambrige,2000.
[18]H. Umegaki, Conditional expectation in an operator algebra Ⅳ, Kodai Math. Sem. Rep.14, (1962),59-85.
[19]O. Klein, Zur quantenmechanischen Begrundung des zweiten Hauptsatzes der Wdrmelehre, Z. Phys.72, (1931),767-775.
[20]H. Araki and E. H. Lieb, Entropy inequalities, Comm. Math. Phys.18, (1970), 160-170.
[21]E. H. Lieb, Convex Trace Functions and the Wigner-Yanase-Dyson Conjecture, Advan. Math.11, (1973),267-288.
[22]E. H. Lieb and M. B. Ruskai, A fundamental property of quantum-mechanical entropy, Phys. Rev. Lett.30, (1973),434-436.
[23]E. H. Lieb and M. B. Ruskai, Proof of the strong subadditivity of quantum mechanical entropy, J. Math. Phys.14, (1973),1938-1941.
[24]A. Wehrl, General properties of entropy, Rev. Mod. Phys.50, (1978),221-260.
[25]H. Araki, Relative entropy of state of von Neumann algebras, Publ. RIMS Kyoto Univ.9, (1976),809-833.
[26]E. Carlen and E. Lieb, A Minkowski type trace inequality and strong subaddi-tivity of quantum entropy, Am. Math. Soc. Trans.189, (1999),59-62.
[27]H. Epstein, Remarks on two theorems of E. Lieb, Commun. Math. Phys.31, (1973),317-325.
[28]E. H. Lieb, Some convexity and subadditivity properties of entropy, Bull. Am. Math. Soc.81, (1975),1-13.
[29]M. Ohya, D. Petz and N. Watanabe, On capacities of quantum channels, Prob. Math. Stats.17, (1997),170-196.
[30]D. Petz, Quasi-entropies for finite quantum systems, Rep. Math. Phys.23, (1986),57-65.
[31]D. Petz, Sufficient subalgebras and the relative entropy, Commun. Math. Phys. 105, (1986),123-131.
[32]D. Petz, A variational expression for the relative entropy, Commun. Math. Phys.114, (1988),345-349.
[33]A. Uhlmann, Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory, Commun. Math. Phys.54, (1971),21-32.
[34]K. Kraus, Effects and Operations:Fundamental Notions of Quantum Theory, Springer-Verlag, Berlin,1983.
[35]C. W. Gardiner, Quantum Noise, Cambrige University Press, Cambrige,1992.
[36]V. B. Braginsky and F. Y. Khahili, Quantum Measurement, Springer-Verlag, Berlin,1991.
[37]A. Peres, How to differentiate between non-orthogonal states, Phys. Lett. A. 128, (1988),19.
[38]L. M. Duan and G. C. Guo, Probabilistic cloning and identification of linearly independent quantum states, Phys. Rev. Lett.80, (1998),4999-5002.
[39]D. J. Foulis and M. K. Bennett, Effect algebras and unsharp quantum logics, Found. Phys.24, (1994),1331-1352.
[40]P. Busch, M. Grabowski and P. J. Lahti, Operational Quantum Physics, Springer-Verlag, Berlin,1995.
[41]S. Gudder and G. Nagy, Sequential quantum measurements, J. Math. Phys.42, (2001),5212-5222.
[42]S. Gudder and R. Greechie, Sequential products on effect algebras, Rep. Math. Phys.49, (2002),87-111.
[43]J. Shen and J. D. Wu, Not each sequential effect algebra is sharply dominating, Phys. Letter A.373, (2009),1708-1712.
[44]J. Shen and J. D. Wu, Remarks on the sequential effect algebras, Report. Math. Phys.63, (2009),441-446.
[45]J. Shen and J. D. Wu, Sequential product on standard effect algebra ε(H), J. Phys. A:Math. Theor.42, (2009),345203-345214.
[46]W. H. Liu and J. D. Wu, A uniqueness problem of the sequence product on operator effect algebra ε(H), J. Phys. A:Math. Theor.42, (2009),185206-185215.
[47]P. Busch, P. J. Lahti and P. Mittlestaedt, The Quantum Theory of Measure-ments, Springer-Verlag, Berlin,1991
[48]P. Busch and J. Singh, Liiders theorem for unsharp quantum measurements, Phys. Lett. A.249, (1998),10-12.
[49]A. Arias, A. Gheondea and S. Gudder, Fixed points of quantum operations, J. Math. Phys.43, (2002),5872-5881.
[50]P. Busch, M. Grabowski and P. J. Lahti, Operational Quantum Physics, Springer-Verlag, Beijing,1999.
[51]P. N. Rathie, Unified (r,s)-entropy and its bivariate measures, Inf. Sci.54, (1991),23-39.
A. Renyi, On measures of entropy and information, Univ. of California Press, Berkeley,1961.
[53]J. Havrda and F. Charvat, Quantification method of classification processes: Concept of structural a-entropy, Kybernetika,3, (1967),30-35.
[54]S. Arimoto, Information-theoretic consideration on estimation problems, In-form, and Control,19, (1971),181-194.
[55]E. K. Lenzi, R. S. Mendes and L. R. da Silva, Statistical mechanics based on Renyi entropy, Phys. A 280, (2000),337-345.
[56]P. Jizba and T. Arimitsu, Observability of Renyi's entropy, Phys. Rev. E 69, (2004),026128.
[57]A. G. Bashkirov, Renyi entropy as a statistical entropy for complex systems, Theor. Math. Phy.149, (2006),1559-1573.
[58]D. R. John, H-theorems for the Tsallis and Renyi entropies, Phys. Lett. A 175, (1993),169-170.
[59]M. M. Ananias, On the irreversible nature of the Tsallis and Renyi entropies, Phys. Lett. A 165, (1992),409-411.
[60]C. Tsallis, R. S. Menders and A. R. Plastino, The role of constraints within generalized nonextensive statistics, Phys. A 261, (1998),534-554.
[61]S. Abe, Axioms and uniqueness theorem for Tsallis entropy, Phys. Lett. A 271, (2000),74-79.
[62]S. Furuichi, Information theoretical properties of Tsallis entropies, J. Math. Phys.47, (2006),023302.
[63]X. H. Hu and Z. X. Ye, Generalized quantum entropy, J. Math. Phys.47, (2006),023502.
[64]K. Yanagi, K. Kuriyama and S. Furuichi, Generalized Shannon inequalities based on Tsallis relative operator entropy, linear Algebr. Appl.394, (2005), 109-118.
[65]S. Abe, Monotonic decrease of the quantum nonadditive divergence by projective measurements, Phys. Lett. A 312, (2003),336-338.
[66]S. Abe and A. K. Rajagopal, Nonadditive conditional entropy and its signifi-cance for local realism, Phys. A 289, (2001),157-164.
[67]S. Furuichi, K. Yanagi and K. Kuriyama, Fundamental properties of Tsallis relative entropy, J. Math. Phys.45, (2004),4868-4877.
[68]S. Furuichi, A note on a parametrically extended entanglement-measure due to Tsallis relative entropy, Info.9, (2006),837-844.
[69]I. J. Taneja, L. Pardo, D. Morales and M. L. Menendez, On generalized information and divergence measures and their applications:a brief review, QUEST11O,13, (1989),47-73.
[70]G. Lindblad, Expectations and Entropy Inequalities for Finite Quantum Sys-tems, Commun. math. Phys.39, (1974),111-119.
[71]G. Lindblad, Completely Positive Maps and Entropy Inequalities, Commun. math. Phys.40, (1975),147-151.
[72]M. B. Ruskai, Inequalities for quantum entropy:A review with conditions for equality, J. Math. Phys.43, (2002),4358-4375.
[73]R. G. Douglas, On majorization and range inclusion of operators in Hilbert space, Proc. Amer. Math. Soc.17, (1966),413-416.
[74]W. F. Stinespring, Positive functions on C*-algebras, Amer. Math. Soc.6, (1955),211-216.
[75]M. D. Choi, Completely positive linear maps on complex matrices, Linear Al-gebr. Appl.10, (1975),285-290.
[76]M. B. Ruskai and F. M. Stillinger, Convexity inequalities for estimating free energy and relative entropy, J. Phys. A 23, (1990),2421-2437.
[77]C. H. Bennett, D. P. Divincenzo, J. A. Smolin and W. K. Wootters, Mixed-state entanglement and quantum error correction, Phys. Rev. A 54, (1996), 3824-3851.
[78]V. Vedral, M. B. Plenio, M. A. Rippin and P. L. Night, Quantifying entangle-ment, Phys. Rev. Lett.78, (1997),2275-2279.
[79]E. Schrodinger and G. Die, Situation in der quantenmechanik, Naturwis-senschaften,23, (1935),807-849.
[80]A. Einstein, B. Podolski and N. Rosen, Can quantum mechanical description of physical reality be considered complete?, Phys. Rev.47, (1935),777-780.
[81]V. Vedral, and M. B. Plenio, Entanglement measures and purification proce-dures, Phys. Rev. A.57, (1998),1619-1633.
[82]M. Horodecki, P. Horodecki and R. Horodecki, Separability of mixed states: necessary and sufficient conditions, Phys. Rev. A.223, (1996),1-8.
[83]M. Horodecki, P. Horodecki and R. Horodecki, Mixed-state entanglement and distillation:is there a'bound'entanglement in nature?, Phys. Rev. Lett.80, (1998),5239-5242.
[84]M. Horodecki, P. Horodecki and R. Horodecki, General teleportation channel, singlet fraction, and quasidistillation, Phys. Rev. A.60, (1999),1888-1898.
[85| M. Horodecki, P. Horodecki and R. Horodecki, Bound entanglement can be activated, Phys. Rev. Lett.82, (1999),1056-1059.
[86]M. Horodecki, P. Horodecki and R. Horodecki, Limits for Entanglement Mea-sures, Phys. Rev. Lett.84, (2000),2014-2017.
[87]E. M. Rains, Bound on distillable entanglement, Phys. Rev. A.60, (1999), 179-184.
[88]G. Vidal, Entanglement monotones, J. Mod. Optic.47, (2000),355-376.
[89]G. Vidal, Entanglement of Pure States for a Single Copy, Phys. Rev. Lett.83, (1999),1046-1049.
[90]A. M. David and R. W. Nolan, Global entanglement in multiparticle systems, J. Math. Phys.43,(2002),4273-4278.
[91]A. Ekert, Quantum cryptography based on Bell's theorem, Phys. Rev. Lett.67, (1991),661-663.
[92]C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Woot-ters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett.70, (1993),1895-1899.
[93]C. H. Bennett and S. J. Wiesner, Communication via one-and two-particle operators on Einstein-Podolsky-Rosen states, Phys. Rev. Lett.69, (1992),2881-2884.
[94]N. J. Cerf and C. Adami, Negative Entropy and Information in Quantum Me-chanics, Phys. Rev. Lett.79, (1997),5194-5197.
[95]R. Horodecki and M. Horodecki, Information-theoretic aspects of inseparability of mixed states, Phys. Rev. A 54, (1996),1838-1843.
[96]K. G. H. Vollbrecht and M. M. Wolf, Conditional entropies and their relation to entanglement criteria, J. Math. Phys.43, (2002),4299-4306.
[97]S. Abe and A. K. Rajagopal, Nonadditive conditional entropy and its signifi-cance for local realism, Phys. A 289, (2001),157-164.
[98]S. Abe and A. K. Rajagopal, Quantum entanglement inferred by the principle of maximum nonadditive entropy, Phys. Rev. A 60, (1999),3461 3466.
[99]E. F. Galvao, M. B. Plenio and S. Virman, Tripartite entenglement and quan-tum relative entropy, J. Phys. A:Math. Gen.33, (2000),8809-8818.
[2]汤苏野.熵：一个世纪之谜的解析(第2版).中国科学技术大学出版社,书号：9787312021268,合肥,2008.
[3]M. Ohya and D. Petz, Quantum Entropy and its Use, Springer-Verlag, Berlin, 1991.
[4]H. J. Yuan, Entropy of Partitions on Quantum Logic, Commun. Theor. Phys. 43, (2005),437-439.
[5]J. L. Kelly, General Topology, Springer-Verlag, New York,1955.
[6]G. Birkhoff and J. von Neumann, The logic of quantum mechanics, Ann. Math. 37, (1936),823-834.
[7]A. Dvurecenskij and S. Pulmannova, New Trends in Quantum Structures, Kluwer,2002.
[8]李丞祖.量子通信和量子计算.国防科技大学出版社,书号：604577,长沙,2001.
[9]S. Kullback and R. A. Leibler. On information and sufficiency, Ann. Math. Stat.22, (1951),79-86.
[10]T. M. Cover and J. A. Thomas, Elements of Information Theory, Wiley, New York,1991
[11]钱伯出.量子力学.电子工业出版社,书号：7-5053-1934-5,北京,1993.
[12]曾谨严.量子力学(卷Ⅰ).第四版,科学出版社,书号：9787030181398,北京,2007.
[13]曾谨严.量子力学(卷Ⅱ),第四版,科学出版社,书号：9787030190215,北京,2007.
[14]周世勋.量子力学教程.高等教育出版社,书号：7040013223,北京,2008.
[15]P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford University Press, London,1958.
[16]L. D. Landau, Non-relativistic Theory, Pergamon, London,1977.
[17]M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Infor-mation, Cambrige University Press, Cambrige,2000.
[18]H. Umegaki, Conditional expectation in an operator algebra Ⅳ, Kodai Math. Sem. Rep.14, (1962),59-85.
[19]O. Klein, Zur quantenmechanischen Begrundung des zweiten Hauptsatzes der Wdrmelehre, Z. Phys.72, (1931),767-775.
[20]H. Araki and E. H. Lieb, Entropy inequalities, Comm. Math. Phys.18, (1970), 160-170.
[21]E. H. Lieb, Convex Trace Functions and the Wigner-Yanase-Dyson Conjecture, Advan. Math.11, (1973),267-288.
[22]E. H. Lieb and M. B. Ruskai, A fundamental property of quantum-mechanical entropy, Phys. Rev. Lett.30, (1973),434-436.
[23]E. H. Lieb and M. B. Ruskai, Proof of the strong subadditivity of quantum mechanical entropy, J. Math. Phys.14, (1973),1938-1941.
[24]A. Wehrl, General properties of entropy, Rev. Mod. Phys.50, (1978),221-260.
[25]H. Araki, Relative entropy of state of von Neumann algebras, Publ. RIMS Kyoto Univ.9, (1976),809-833.
[26]E. Carlen and E. Lieb, A Minkowski type trace inequality and strong subaddi-tivity of quantum entropy, Am. Math. Soc. Trans.189, (1999),59-62.
[27]H. Epstein, Remarks on two theorems of E. Lieb, Commun. Math. Phys.31, (1973),317-325.
[28]E. H. Lieb, Some convexity and subadditivity properties of entropy, Bull. Am. Math. Soc.81, (1975),1-13.
[29]M. Ohya, D. Petz and N. Watanabe, On capacities of quantum channels, Prob. Math. Stats.17, (1997),170-196.
[30]D. Petz, Quasi-entropies for finite quantum systems, Rep. Math. Phys.23, (1986),57-65.
[31]D. Petz, Sufficient subalgebras and the relative entropy, Commun. Math. Phys. 105, (1986),123-131.
[32]D. Petz, A variational expression for the relative entropy, Commun. Math. Phys.114, (1988),345-349.
[33]A. Uhlmann, Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory, Commun. Math. Phys.54, (1971),21-32.
[34]K. Kraus, Effects and Operations:Fundamental Notions of Quantum Theory, Springer-Verlag, Berlin,1983.
[35]C. W. Gardiner, Quantum Noise, Cambrige University Press, Cambrige,1992.
[36]V. B. Braginsky and F. Y. Khahili, Quantum Measurement, Springer-Verlag, Berlin,1991.
[37]A. Peres, How to differentiate between non-orthogonal states, Phys. Lett. A. 128, (1988),19.
[38]L. M. Duan and G. C. Guo, Probabilistic cloning and identification of linearly independent quantum states, Phys. Rev. Lett.80, (1998),4999-5002.
[39]D. J. Foulis and M. K. Bennett, Effect algebras and unsharp quantum logics, Found. Phys.24, (1994),1331-1352.
[40]P. Busch, M. Grabowski and P. J. Lahti, Operational Quantum Physics, Springer-Verlag, Berlin,1995.
[41]S. Gudder and G. Nagy, Sequential quantum measurements, J. Math. Phys.42, (2001),5212-5222.
[42]S. Gudder and R. Greechie, Sequential products on effect algebras, Rep. Math. Phys.49, (2002),87-111.
[43]J. Shen and J. D. Wu, Not each sequential effect algebra is sharply dominating, Phys. Letter A.373, (2009),1708-1712.
[44]J. Shen and J. D. Wu, Remarks on the sequential effect algebras, Report. Math. Phys.63, (2009),441-446.
[45]J. Shen and J. D. Wu, Sequential product on standard effect algebra ε(H), J. Phys. A:Math. Theor.42, (2009),345203-345214.
[46]W. H. Liu and J. D. Wu, A uniqueness problem of the sequence product on operator effect algebra ε(H), J. Phys. A:Math. Theor.42, (2009),185206-185215.
[47]P. Busch, P. J. Lahti and P. Mittlestaedt, The Quantum Theory of Measure-ments, Springer-Verlag, Berlin,1991
[48]P. Busch and J. Singh, Liiders theorem for unsharp quantum measurements, Phys. Lett. A.249, (1998),10-12.
[49]A. Arias, A. Gheondea and S. Gudder, Fixed points of quantum operations, J. Math. Phys.43, (2002),5872-5881.
[50]P. Busch, M. Grabowski and P. J. Lahti, Operational Quantum Physics, Springer-Verlag, Beijing,1999.
[51]P. N. Rathie, Unified (r,s)-entropy and its bivariate measures, Inf. Sci.54, (1991),23-39.
A. Renyi, On measures of entropy and information, Univ. of California Press, Berkeley,1961.
[53]J. Havrda and F. Charvat, Quantification method of classification processes: Concept of structural a-entropy, Kybernetika,3, (1967),30-35.
[54]S. Arimoto, Information-theoretic consideration on estimation problems, In-form, and Control,19, (1971),181-194.
[55]E. K. Lenzi, R. S. Mendes and L. R. da Silva, Statistical mechanics based on Renyi entropy, Phys. A 280, (2000),337-345.
[56]P. Jizba and T. Arimitsu, Observability of Renyi's entropy, Phys. Rev. E 69, (2004),026128.
[57]A. G. Bashkirov, Renyi entropy as a statistical entropy for complex systems, Theor. Math. Phy.149, (2006),1559-1573.
[58]D. R. John, H-theorems for the Tsallis and Renyi entropies, Phys. Lett. A 175, (1993),169-170.
[59]M. M. Ananias, On the irreversible nature of the Tsallis and Renyi entropies, Phys. Lett. A 165, (1992),409-411.
[60]C. Tsallis, R. S. Menders and A. R. Plastino, The role of constraints within generalized nonextensive statistics, Phys. A 261, (1998),534-554.
[61]S. Abe, Axioms and uniqueness theorem for Tsallis entropy, Phys. Lett. A 271, (2000),74-79.
[62]S. Furuichi, Information theoretical properties of Tsallis entropies, J. Math. Phys.47, (2006),023302.
[63]X. H. Hu and Z. X. Ye, Generalized quantum entropy, J. Math. Phys.47, (2006),023502.
[64]K. Yanagi, K. Kuriyama and S. Furuichi, Generalized Shannon inequalities based on Tsallis relative operator entropy, linear Algebr. Appl.394, (2005), 109-118.
[65]S. Abe, Monotonic decrease of the quantum nonadditive divergence by projective measurements, Phys. Lett. A 312, (2003),336-338.
[66]S. Abe and A. K. Rajagopal, Nonadditive conditional entropy and its signifi-cance for local realism, Phys. A 289, (2001),157-164.
[67]S. Furuichi, K. Yanagi and K. Kuriyama, Fundamental properties of Tsallis relative entropy, J. Math. Phys.45, (2004),4868-4877.
[68]S. Furuichi, A note on a parametrically extended entanglement-measure due to Tsallis relative entropy, Info.9, (2006),837-844.
[69]I. J. Taneja, L. Pardo, D. Morales and M. L. Menendez, On generalized information and divergence measures and their applications:a brief review, QUEST11O,13, (1989),47-73.
[70]G. Lindblad, Expectations and Entropy Inequalities for Finite Quantum Sys-tems, Commun. math. Phys.39, (1974),111-119.
[71]G. Lindblad, Completely Positive Maps and Entropy Inequalities, Commun. math. Phys.40, (1975),147-151.
[72]M. B. Ruskai, Inequalities for quantum entropy:A review with conditions for equality, J. Math. Phys.43, (2002),4358-4375.
[73]R. G. Douglas, On majorization and range inclusion of operators in Hilbert space, Proc. Amer. Math. Soc.17, (1966),413-416.
[74]W. F. Stinespring, Positive functions on C*-algebras, Amer. Math. Soc.6, (1955),211-216.
[75]M. D. Choi, Completely positive linear maps on complex matrices, Linear Al-gebr. Appl.10, (1975),285-290.
[76]M. B. Ruskai and F. M. Stillinger, Convexity inequalities for estimating free energy and relative entropy, J. Phys. A 23, (1990),2421-2437.
[77]C. H. Bennett, D. P. Divincenzo, J. A. Smolin and W. K. Wootters, Mixed-state entanglement and quantum error correction, Phys. Rev. A 54, (1996), 3824-3851.
[78]V. Vedral, M. B. Plenio, M. A. Rippin and P. L. Night, Quantifying entangle-ment, Phys. Rev. Lett.78, (1997),2275-2279.
[79]E. Schrodinger and G. Die, Situation in der quantenmechanik, Naturwis-senschaften,23, (1935),807-849.
[80]A. Einstein, B. Podolski and N. Rosen, Can quantum mechanical description of physical reality be considered complete?, Phys. Rev.47, (1935),777-780.
[81]V. Vedral, and M. B. Plenio, Entanglement measures and purification proce-dures, Phys. Rev. A.57, (1998),1619-1633.
[82]M. Horodecki, P. Horodecki and R. Horodecki, Separability of mixed states: necessary and sufficient conditions, Phys. Rev. A.223, (1996),1-8.
[83]M. Horodecki, P. Horodecki and R. Horodecki, Mixed-state entanglement and distillation:is there a'bound'entanglement in nature?, Phys. Rev. Lett.80, (1998),5239-5242.
[84]M. Horodecki, P. Horodecki and R. Horodecki, General teleportation channel, singlet fraction, and quasidistillation, Phys. Rev. A.60, (1999),1888-1898.
[85| M. Horodecki, P. Horodecki and R. Horodecki, Bound entanglement can be activated, Phys. Rev. Lett.82, (1999),1056-1059.
[86]M. Horodecki, P. Horodecki and R. Horodecki, Limits for Entanglement Mea-sures, Phys. Rev. Lett.84, (2000),2014-2017.
[87]E. M. Rains, Bound on distillable entanglement, Phys. Rev. A.60, (1999), 179-184.
[88]G. Vidal, Entanglement monotones, J. Mod. Optic.47, (2000),355-376.
[89]G. Vidal, Entanglement of Pure States for a Single Copy, Phys. Rev. Lett.83, (1999),1046-1049.
[90]A. M. David and R. W. Nolan, Global entanglement in multiparticle systems, J. Math. Phys.43,(2002),4273-4278.
[91]A. Ekert, Quantum cryptography based on Bell's theorem, Phys. Rev. Lett.67, (1991),661-663.
[92]C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Woot-ters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett.70, (1993),1895-1899.
[93]C. H. Bennett and S. J. Wiesner, Communication via one-and two-particle operators on Einstein-Podolsky-Rosen states, Phys. Rev. Lett.69, (1992),2881-2884.
[94]N. J. Cerf and C. Adami, Negative Entropy and Information in Quantum Me-chanics, Phys. Rev. Lett.79, (1997),5194-5197.
[95]R. Horodecki and M. Horodecki, Information-theoretic aspects of inseparability of mixed states, Phys. Rev. A 54, (1996),1838-1843.
[96]K. G. H. Vollbrecht and M. M. Wolf, Conditional entropies and their relation to entanglement criteria, J. Math. Phys.43, (2002),4299-4306.
[97]S. Abe and A. K. Rajagopal, Nonadditive conditional entropy and its signifi-cance for local realism, Phys. A 289, (2001),157-164.
[98]S. Abe and A. K. Rajagopal, Quantum entanglement inferred by the principle of maximum nonadditive entropy, Phys. Rev. A 60, (1999),3461 3466.
[99]E. F. Galvao, M. B. Plenio and S. Virman, Tripartite entenglement and quan-tum relative entropy, J. Phys. A:Math. Gen.33, (2000),8809-8818.