Some Properties And Applications Of Entropy And Information

The concept of entropy has its origin in thermodynamics,and is fundamental inquantifying randomness and uncertainties of states.Entropy and its various derivativessuch as relative entropy and mutual information play an important role in both classicaland quantum information theory.In this paper,some properties and applications ofgeneralized entropy,quantum relative entropy and quantum information are studied.The stability of entropy is a basic and important issue.In Chapter 2,the stabilitiesof both classical and quantum generalized entropy are discussed for finite systems.Byuse of a probabilistic coupling technique introduced by Zhang,several sharp inequalitiesconcerning the uniform continuity of some generalized entropies have been obtained,which generalize some existing results such as the stabilities of Shannon entropy,vonNeumann entropy and a class of nonaddtive entropies(e.g.,Tsallis entropy).In Chapter 3,several notions of relative entropy between quantum ensembles areproposed from two different perspectives.The first is based on a probabilistic couplingtechnique,and the second is a measurement-based approach.Both lead to naturalgeneralizations of the relative entropy between quantum states.Using the notion ofrelative entropy between quantum ensembles,the quantumness and distinguishability ofquantum ensembles are quantified.By the notion of relative entropy between quantumensembles,it is demonstrated that a set consisting of two pure states is the most quantumif and only if the states are 45°apart.Quantum mutual information is a particular instance of quantum relative entropy,and is usually regarded as a measure of total correlations in a bipartite state.This notionhas important physical meaning and subtle mathematical properties.In Chapter(?),(?)basic properties of quantum mutual information are reviewed for the purpose of the nexttwo chapters.Moreover,several measures of quantum correlation are introduced andevaluated explicitly for some important states.Using the notion of quantum mutual information,the decoherence of quantumoperation is discussed in Chapter 5.A notion of decoherent information of quantumoperation is proposed,and its basic properties are investigated.Decoherent informationof several important quantum operations are calculated explitly.A continuity estimatefor the decoherent information of quantum operation is established.A no-broadcasting theorem for quantum information is established by virtue of the decoherent informationof quantum operation.No-broadcasting theorems are important results in quantum information theorywith profound applications.In Chapter 6,using the monotonicity of quantum relativeentropy,the relation between the no-broadcasting theorem for quantum states estab-lished by Barnum et al.and the no-local-broadcasting theorem for quantum correla-tions proposed by Piani et al.is discussed.It is conjectured that the two differentno-broadcasting theorems are equivalent.It is proved that the conjecture holds under acondition of non-degenerate spectrum.