Entropy, Divergence, and Majorization in Classical and Quantum Thermodynamics

The first step in quantum information theory is the identification of entanglement as a valuable resource. The next step is learning how to exploit this resource efficiently. We learn how to exploit entanglement efficiently by applying analogues of thermodynamical concepts. These concepts include reversibility, entropy, and the distinction between intensive and extensive quantities. We discuss some of these analogues and show how they lead to a measure of entanglement for pur...

This is a review on entropy in various fields of mathematics and science. Its scope is to convey a unified vision of the classical as well as some newer entropy notions to a broad audience with an intermediate background in dynamical systems and ergodic theory. Due to the breadth and depth of the subject, we have opted for a compact exposition whose contents are a compromise between conceptual import and instrumental relevance. The intended technical level and the space limit...

Entanglement is central both to the foundations of quantum theory and, as a novel resource, to quantum information science. The theory of entanglement establishes basic laws, such as the non-increase of entanglement under local operations, that govern its manipulation and aims to draw from them formal analogies to the second law of thermodynamics. However, while in the second law the entropy uniquely determines whether a state is adiabatically accessible from another, the man...

One of the predominant challenges when engineering future quantum information processors is that large quantum systems are notoriously hard to maintain and control accurately. It is therefore of immediate practical relevance to investigate quantum information processing with limited physical resources, for example to ask: How well can we perform information processing tasks if we only have access to a small quantum device? Can we beat fundamental limits imposed on information...

This book provides an introduction to the emerging field of quantum thermodynamics, with particular focus on its relation to quantum information and its implications for quantum computers and next generation quantum technologies. The text, aimed at graduate level physics students with a working knowledge of quantum mechanics and statistical physics, provides a brief overview of the development of classical thermodynamics and its quantum formulation in Chapter 1. Chapter 2 the...

Even today, the concept of entropy is perceived by many as quite obscure. The main difficulty is analyzed as being fundamentally due to the subjectivity and anthropocentrism of the concept that prevent us to have a sufficient distance to embrace it. However, it is pointed out that the lack of coherence of certain presentations or certain preconceived ideas do not help. They are of three kinds : 1) axiomatic thermodynamics; 2) inconsistent solutions of certain paradoxes; 3) re...

The theory of majorization and its variants, including thermomajorization, have been found to play a central role in the formulation of many physical resource theories, ranging from entanglement theory to quantum thermodynamics. Here we formulate the framework of quantum relative Lorenz curves, and show how it is able to unify majorization, thermomajorization, and their noncommutative analogues. In doing so, we define the family of Hilbert $\alpha$-divergences and show how it...

Several recent results on thermodynamics have been obtained using the tools of quantum information theory and resource theories. So far, the resource theories utilised to describe thermodynamics have assumed the existence of an infinite thermal reservoir, by declaring that thermal states at some background temperature come for free. Here, we propose a resource theory of quantum thermodynamics without a background temperature, so that no states at all come for free. We apply t...

Information theory is a mathematical theory of learning with deep connections with topics as diverse as artificial intelligence, statistical physics, and biological evolution. Many primers on information theory paint a broad picture with relatively little mathematical sophistication, while many others develop specific application areas in detail. In contrast, these informal notes aim to outline some elements of the information-theoretic "way of thinking," by cutting a rapid a...

Thermodynamics is based on the notions of energy and entropy. While energy is the elementary quantity governing physical dynamics, entropy is the fundamental concept in information theory. In this work, starting from first principles, we give a detailed didactic account on the relations between energy and entropy and thus physics and information theory. We show that thermodynamic process inequalities, like the Second Law, are equivalent to the requirement that an effective de...