We review the concept of majorization and its relation with generalized information measures. Majorization theory provides an elegant framework for comparing two probability distributions, leading to a rigorous concept of disorder which is more stringent than that based on the Shannon entropy. Nevertheless, it is shown that it can be fully captured through general entropic inequalities based on generalized entropic forms. A brief review of generalized entropies is also provided. As illustration, we discuss the majorization properties of generalized thermal distributions derived from generalized entropies, and identify rigorous mixing parameters. We also describe majorization in quantum systems. We discuss in particular its capability for providing a disorder based criterion for the detection of quantum entanglement, which is stronger than that based on the von Neumann entropy and leads to a generalized entropic separability criterion.
We review the concept of majorization and its relation with generalized information measures. Majorization theory provides an elegant framework for comparing two probability distributions, leading to a rigorous concept of disorder which is more stringent than that based on the Shannon entropy. Nevertheless, it is shown that it can be fully captured through general entropic inequalities based on generalized entropic forms. A brief review of generalized entropies is also provided. As illustration, we discuss the majorization properties of generalized thermal distributions derived from generalized entropies, and identify rigorous mixing parameters. We also describe majorization in quantum systems. We discuss in particular its capability for providing a disorder based criterion for the detection of quantum entanglement, which is stronger than that based on the von Neumann entropy and leads to a generalized entropic separability criterion.
The theory of majorization is a powerful mathematical tool which naturally arises in the quantum theory as a consequence of its fundamental connection with unitarity. In this work, we look for majorization relations based on entropic inequalities. We begin by introducing the theory of majorization. We then give some notions of quantum mechanics, like the density matrix and some algebra, before presenting the concept of entropy which is closely related to the theory of majorization. We define in the process the Shannon, von Neumann and Renyi entropies, along with some of their properties and applications. We then define quantum entanglement and give some criteria for separability, which turn out to be applications of the concepts of majorization and entropy in the quantum theory. We present some new majorization relations similar to already existing entropic inequalities, and exhibit their advantages and limitations. The rest of the report is dedicated to the study of the field of Gaussian quantum information. We present the theory, before introducing other majorization relations, which turn out to be useful because they allow one to know if it is possible to transform one quantum state into another when there exists majorization relations between them.
We improve the entropic uncertainty relations for position and momentum coarse-grained measurements. We derive the continuous, coarse-grained counterparts of the discrete uncertainty relations based on the concept of majorization. The entropic inequalities obtained involve two R\'enyi entropies of the same order, and thus go beyond the standard scenario with conjugated parameters. In a special case describing the sum of two Shannon entropies, the majorization-based bounds significantly outperform the currently known results in the regime of larger coarse graining, and might thus be useful for entanglement detection in continuous variables.
Uncertainty principle, which was first introduced by Werner Heisenberg in 1927, forms a fundamental component of quantum mechanics. A graceful aspect of quantum mechanics is that the uncertainty relations between incompatible observables allow for succinct quan- titative formulations of this revolutionary idea: it is impossible to simultaneously measure two complementary variables of a particle in precision. In particular, information theory offers two basic ways to express the Heisenberg’s principle: variance-based uncertainty relations and entropic uncertainty relations. We first investigate the uncertainty relations based on the sum of variances and derive a family of weighted uncertainty relations to provide an optimal lower bound for all situations. Our work indicates that it seems unreasonable to assume a priori that incompatible observables have equal contribution to the variance-based sum form uncertainty relations. We also study the role of mutually exclusive physical states in the recent work and generalize the variance-based uncertainty relations to mutually exclusive uncertainty relations. Next, we develop a new kind of entanglement detection criteria within the framework of marjorization theory and its matrix representation. By virtue of majorization uncertainty bounds, we are able to construct the entanglement criteria which have advantage over the scalar detect- ing algorithms as they are often stronger and tighter. Furthermore, we explore various expression of entropic uncertainty relations, including sum of Shannon entropies, majorization uncer- tainty relations and uncertainty relations in presence of quantum memory. For entropic uncertainty relations without quantum side information, we provide several tighter bounds for multi-measurements, with some of them also valid for Rényi and Tsallis entropies besides the Shannon entropy. We employ majorization theory and actions of the symmetric group to obtain an admixture bound for entropic uncertainty relations with multi-measurements. Comparisons among existing bounds for multi-measurements are also given. However,classical entropic uncertainty relations assume there has only classical side information. For modern uncertainty relations, those who allowed for non-trivial amount of quantum side information, their bounds have been strengthened by our recent result for both two and multi- measurements. Finally, we propose an approach which can extend all uncertainty relations on Shannon entropies to allow for quantum side information and discuss the applications of our entropic framework. Combined with our uniform entanglement frames, it is possible to detect entanglement via entropic uncertainty relations even if there is no quantum side in- formation. With the rising of quantum information theory, uncertainty relations have been established as important tools for a wide range of applications, such as quantum cryptography, quantum key distribution, entanglement detection, quantum metrology, quantum speed limit and so on. It is thus necessary to focus on the study of uncertainty relations.
In this paper, we discuss some general connections between the notions of positive map, weak majorization and entropic inequalities in the context of detection of entanglement among bipartite quantum systems. First, basing on the fact that any positive map can be written as the difference between two completely positive maps Λ=Λ 1 −Λ 2 , we propose a possible way to generalize the Nielsen–Kempe majorization criterion. Then, we present two methods of derivation of some general classes of entropic inequalities useful for the detection of entanglement. While the first one follows from the aforementioned generalized majorization relation and the concept of Schur-concave decreasing functions, the second is based on some functional inequalities. What is important is that, contrary to the Nielsen–Kempe majorization criterion and entropic inequalities, our criteria allow for the detection of entangled states with positive partial transposition when using indecomposable positive maps. We also point out that if a state with at least one maximally mixed subsystem is detected by some necessary criterion based on the positive map Λ, then there exist entropic inequalities derived from Λ (by both procedures) that also detect this state. In this sense, they are equivalent to the necessary criterion [ I ⊗Λ](ϱ AB )⩾0. Moreover, our inequalities provide a way of constructing multi-copy entanglement witnesses and therefore are promising from the experimental point of view. Finally, we discuss some of the derived inequalities in the context of the recently introduced protocol of state merging and the possibility of approximating the mean value of a linear entanglement witness.
Entropies play a fundamental role in classical and quantum information theory. There are a plethora of quantum entropies and they satisfy a host of interesting properties. The von Neumann entropy, the first quantum entropy to be defined and arguably the most important of them, satisfies a powerful property called strong subadditivity. It was proved by Lieb and Ruskai in 1973 and has turned out to be an essential tool in establishing various important results in quantum information theory. We briefly review this property and its applications. Our main focus, however, is on another desirable property of entropies, namely, continuity. We introduce a notion of majorization flow and demonstrate it to be a powerful tool for deriving simple and universal proofs of continuity bounds for Schur concave functions with a particular emphasis on entropic functions relevant in information theory. In particular, for the case of the α-Rényi entropy, whose connections to thermodynamics are discussed in this chapter, majorization flow yields a Lipschitz continuity bound for the case α>1, thus resolving an open problem and providing a substantial improvement over previously known bounds.
The Rényi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies, or mutual information, and have found many applications in information theory and beyond. Various generalizations of Rényi entropies to the quantum setting have been proposed, most prominently Petz's quasi-entropies and Renner's conditional min-, max-, and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Rényi entropies that contains the von Neumann entropy, min-entropy, collision entropy, and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.