Trace inequalities and quantum entropy: an introductory course

Following ref [1], a classical upper bound for quantum entropy is identified and illustrated, 0 ≤ Sq ≤ ln(eσ2/ 2􏰁), involving the variance σ2 in phase space of the classical limit distribution of a given system. A fortiori, this further bounds the corresponding information-theoretical generalizations of the quantum entropy proposed by Re ́nyi.

Linear Algebra and its Applications, 2005

Certain trace inequalities related to matrix logarithm are shown. These results enable us to give a partial answer of the open problem conjectured by A.S.Holevo. That is, concavity of the auxiliary function which appears in the random coding exponent as the lower bound of the quantum reliability function for general quantum states is proven in the case of 0 ≤ s ≤ 1.

Journal of Physics A: Mathematical and General, 2003

We show how to determine the maximum and minimum possible values of one measure of entropy for a given value of another measure of entropy. These maximum and minimum values are obtained for two standard forms of probability distribution (or quantum state) independent of the entropy measures, provided the entropy measures satisfy a concavity/convexity relation. These results may be applied to entropies for classical probability distributions, entropies of mixed quantum states and measures of entanglement for pure states.

Bell and Bell-like inequalities are essential tools for identifying and inspecting quantum entanglement, a key phenomenon in quantum information theory. As the development of Quantum Information Theory progresses more and more Bell-like inequalities are formulated. An investigation of these and further inequalities will be given in their correspondences to Bell’s and Kochen-Specker theorems. Also Hardy’s test will be studied and particular applications of it for bipartite and tripartite systems are going to be probed

Journal of Physics A: Mathematical and Theoretical

We propose an extension of the quantum entropy power inequality for finite dimensional quantum systems, and prove a conditional quantum entropy power inequality by using the majorization relation as well as the concavity of entropic functions also given by Audenaert, Datta, and Ozols [J. Math. Phys. 57, 052202 (2016)]. Here, we make particular use of the fact that a specific local measurement after a partial swap operation (or partial swap quantum channel) acting only on finite dimensional bipartite subsystems does not affect the majorization relation for the conditional output states when a separable ancillary subsystem is involved. We expect our conditional quantum entropy power inequality to be useful, and applicable in bounding and analyzing several capacity problems for quantum channels.

2006

We study the open problem given by Holevo and Ogawa-Nagaoka on the concavity of the auxiliary func- tion of the quantum reliability function. Firstly we review the previous results on this problem in the case that the parameter s is positive. Secondly we consider the problem in the case that the parameter s is nega- tive.

Colloquium Mathematicum, 2013

In this paper we investigate a notion of relative operator entropy, which develops the theory started by J.I. Fujii and E. Kamei [Math. Japonica 34 (1989), 341-348]. For two finite sequences A = (A 1 , · · · , A n ) and B = (B 1 , · · · , B n ) of positive operators acting on a Hilbert space, a real number q and an operator monotone function f we extend the concept of entropy by S f q (A|B) := 2010 Mathematics Subject Classification. Primary 47A63; Secondary 15A42, 46L05, 47A30.

Mathematics

We consider a probability distribution p0(x),p1(x),… depending on a real parameter x. The associated information potential is S(x):=∑kpk2(x). The Rényi entropy and the Tsallis entropy of order 2 can be expressed as R(x)=−logS(x) and T(x)=1−S(x). We establish recurrence relations, inequalities and bounds for S(x), which lead immediately to similar relations, inequalities and bounds for the two entropies. We show that some sequences Rn(x)n≥0 and Tn(x)n≥0, associated with sequences of classical positive linear operators, are concave and increasing. Two conjectures are formulated involving the information potentials associated with the Durrmeyer density of probability, respectively the Bleimann–Butzer–Hahn probability distribution.

Advances in Theoretical and Mathematical Physics

In these notes we will give an overview and road map for a definition and characterization of (relative) entropy for both classical and quantum systems. In other words, we will provide a consistent treatment of entropy which can be applied within the recently developed Orlicz space based approach to large systems. This means that the proposed approach successfully provides a refined framework for the treatment of entropy in each of classical statistical physics, Dirac's formalism of Quantum Mechanics, large systems of quantum statistical physics, and finally also for Quantum Field Theory. Despite the efforts of many authors over a very long period of time, gaining a deeper understanding of entropy remains one of the most important and intriguing challenges in the physics of large systems -a challenge still receiving the close attention of many prominent authors. See for example . In this endeavour the techniques available for the quantum framework still lack the refinement of those available for classical systems. On this point, Dirac's formalism for Quantum Mechanics and von Neumann's definition of entropy in the context of B(H), does however provide a "template" for developing techniques for the description and study of entropy in the context of tracial von Neumann algebras. One possible way in which von Neumann's ideas could be refined to provide a "good" description of states with well-defined entropy in the tracial case, was fully described in . As is shown in that paper, a successful description of states with entropy can be achieved on condition that the more common framework for quantum theory based on the pair of spaces L ∞ , L 1 , is replaced with a formalism based on the pair of Orlicz spaces L cosh -1 , L log(L + 1) . An important point worth noting (also pointed out in ), is that this axiomatic shift leaves Dirac's formalism intact! However not all quantum systems correspond to tracial von Neumann algebras. (Note for example that the local algebras of Quantum Field Theory are type III 1 .) Hence no formalism for describing and studying entropy is complete, if it cannot also find expression in a type III context. In these notes we will provide a formalism for describing (relative) entropy for the most general quantum systems. Our approach is to describe relative entropy in terms of modular dynamics, for which a common input stemming from the concept of Radon-Nikodym derivatives is crucial. As we

Science China Physics, Mechanics & Astronomy, 2016