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Entropies, majorization flow, and continuity bounds

Eric P. Hanson, Nilanjana Datta

Published Jun 29, 2022• 0 citations
Publisher

Abstract

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 $\alpha$-Rényi entropy, whose connections to thermodynamics are discussed in this chapter, majorization flow yields a Lipschitz continuity bound for the case $\alpha>1$, thus resolving an open problem and providing a substantial improvement over previously known bounds.

DOI
10.4171/90-1/20

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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\alphaα-Rényi entropy, whose connections to thermodynamics are discussed in this chapter, majorization flow yields a Lipschitz continuity bound for the case α>1\alpha>1α>1, thus resolving an open problem and providing a substantial improvement over previously known bounds.
In the first part of this thesis, we present a general technique for establishing local and uniform continuity bounds for Schur concave functions. Our technique uses a particular relationship between majorization and the trace distance between quantum states. Namely, the majorization pre-order attains a minimum over εεε-balls in this distance. By tracing the path of the majorization-minimizer as a function of the distance εεε, we obtain the path of "majorization flow". This yields a new proof of the Audenaert-Fannes continuity bound for the von Neumann entropy in a universal framework which extends to the other functions, including the ααα-Rényi entropy, for which we obtain novel bounds in the case α>1α> 1α>1. We apply this technique to other Schur concave functions, such as the number of connected components of a certain random graph model, and the number of distinct realizations of a random variable. In the second part, we consider repeated interaction systems, in which a system of interest interacts with a sequence of probes one at a time. We characterize which repeated interaction systems break any initially-present entanglement between the system and an untouched reference after finitely many steps. Additionally, when the probes and their interactions with the system are slowly-varying (i.e. adiabatic), we analyze the saturation of Landauer's bound, an inequality between the entropy change of the system and the energy change of the probes, in the limit in which the number of steps tends to infinity and both the difference between consecutive probes and the difference between their interactions vanishes. This analysis proceeds at a fine-grained level by means of a two-time measurement protocol, in which the energy of the probes is measured before and after each interaction.
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.
Abstract Chapter 20 provides a discussion of entropies and entropic quantities in quantum information theory, briefly considering the generalizations of the Rényi entropy to the quantum regime, but mainly focusing on the von Neumann entropy and its properties such as concavity, the Araki-Lieb inequality, and subadditivity. We introduce the quantum relative entropy, quantum joint entropy, quantum conditional entropy, and quantum mutual information, and prove key properties and results for these quantities, including Klein’s inequality, joint convexity, additivity, and monotonicity under completely positive and trace-preserving maps of the relative entropy, as well as weak monotonicity and strong subadditivity of the von Neumann entropy. We then analyse the relation of (negative) conditional entropy and entanglement. In this context we discuss the conditional amplitude operator and the mutual amplitude operator, as well as conditional Rényi entropies and their role for entanglement detection.
Given an arbitrary quantum state (σ), we obtain an explicit construction of a state ρε*(σ) [respectively, ρ*,ε(σ)] which has the maximum (respectively, minimum) entropy among all states which lie in a specified neighborhood (ε-ball) of σ. Computing the entropy of these states leads to a local strengthening of the continuity bound of the von Neumann entropy, i.e., the Audenaert-Fannes inequality. Our bound is local in the sense that it depends on the spectrum of σ. The states ρε*(σ) and ρ*,ε(σ) depend only on the geometry of the ε-ball and are in fact optimizers for a larger class of entropies. These include the Rényi entropy and the minimum- and maximum-entropies, providing explicit formulas for certain smoothed quantities. This allows us to obtain local continuity bounds for these quantities as well. In obtaining this bound, we first derive a more general result which may be of independent interest, namely, a necessary and sufficient condition under which a state maximizes a concave and Gâteaux-differentiable function in an ε-ball around a given state σ. Examples of such a function include the von Neumann entropy and the conditional entropy of bipartite states. Our proofs employ tools from the theory of convex optimization under non-differentiable constraints, in particular Fermat’s rule, and majorization theory.
We present a continuity bound for quantum conditional entropies in terms of fidelity, with no dependence on the dimension of the conditioning system. Our results encompass both the von Neumann entropy and one version of sandwiched Rényi entropy, which is particularly relevant for applications in quantum key distribution (QKD). In such applications, this bound should provide improved results as compared to previous continuity bounds based on trace distance. We also show that the bounds can be sharpened as expected for classical-quantum states, and verify that they hold when the conditioning system has countably infinite dimension. Finally, we present several variants tailored for exponential functions of the Rényi entropies instead, which should be relevant for QKD applications.