Abstract Entropy has played an essential role in the history of physics. Its mathematical definition and applications have changed over time till today. In this paper, we first review the historical evolution of these various points of view, from the thermodynamic definition to information entropy from Shannon in classical physics, up to the modern concept of Neumann’s quantum entropy. As a specific example, we consider entanglement entropy and compare the phase space approach in classical physics to the Hilbert space approach in quantum physics in simple model systems. We derive a general expression for the entanglement entropy of fermions and bosons in arbitrary partitions of Hilbert space, valid beyond the thermodynamic limit. Next, we compare thermodynamic heat engines with quantum heat engines. Finally, we proceed to the more general concept of quantum (computational) complexity and argue, using the concept of entanglement entropy, that the Heisenberg time in classically chaotic systems coincides with the time when maximal complexity is reached in the quantum case for systems with all–all interactions.
Abstract Entropy has played an essential role in the history of physics. Its mathematical definition and applications have changed over time till today. In this paper, we first review the historical evolution of these various points of view, from the thermodynamic definition to information entropy from Shannon in classical physics, up to the modern concept of Neumann’s quantum entropy. As a specific example, we consider entanglement entropy and compare the phase space approach in classical physics to the Hilbert space approach in quantum physics in simple model systems. We derive a general expression for the entanglement entropy of fermions and bosons in arbitrary partitions of Hilbert space, valid beyond the thermodynamic limit. Next, we compare thermodynamic heat engines with quantum heat engines. Finally, we proceed to the more general concept of quantum (computational) complexity and argue, using the concept of entanglement entropy, that the Heisenberg time in classically chaotic systems coincides with the time when maximal complexity is reached in the quantum case for systems with all–all interactions.
Classical formulations of the entropy concept and its interpretation are introduced. This is to motivate the definition of the quantum von Neumann entropy. Some general properties of quantum entropy are developed, such as the quantum entropy which always increases. The current state of the area that includes thermodynamics and quantum mechanics is reviewed. This interaction shall be critical for the development of nonequilibrium thermodynamics. The Jarzynski inequality is developed in two separate but related ways. The nature of irreversibility and its role in physics are considered as well. Finally, a specific quantum spin model is defined and is studied in such a way as to illustrate many of the subjects that have appeared.
Entropy is a fundamental concept which emerged along with other ideas during the development of thermodynamics and statistical mechanics Landau and Lifshitz (1978); Lieb and Yngvason (1999). Entropy has developed foremost out of phenomenological thermodynamical considerations such as the second law of thermodynamics in which it plays a prominent role Wehrl (1978). With the intense interest in the investigation of the physics of matter at the atomic and subatomic quantum levels, it maywell be askedwhether this concept can emerge out of the study of systems at a more fundamental level. In fact, it may be argued that a correct definition is only possible in the framework of quantum mechanics, whereas in classical mechanics, entropy can only be introduced in a rather limited and artificial way. Entropy relates macroscopic and microscopic aspects of nature, and ultimately determines the behavior of macroscopic systems. It is the intention here to present an introduction to this subject in a readable manner from the quantum point of view. There are many reasons for undertaking this. The intense interest in irreversible thermodynamics Grossing (2008), the statistical mechanics of astrophysical objects Padmanabhan (1990); Pathria (1977), quantum gravity and entropy of black holes Peres & al. (2004), testing quantum mechanics Ballentine (1970) and applications to condensed matter and quantum optics Haroche & al. (2006); Raimond & al. (2001) are just a few areas which are directly or indirectly touched on here. Let us begin by introducing the concept of entropy from the quantummechanical perspective, realizing that the purpose is to focus on quantum mechanics in particular. Quantum mechanics makes a clear distinction between observables and states. Observables such as position and momentum are mathematically described by self-adjoint operators in a Hilbert space. States, which are generally mixed, can be described by a density matrix, which is designated by ρ throughout. This operator ρ is Hermitean, has trace one and yields the expectation value of an observable A in the state ρ through the definition
Entropy is one of the most basic concepts in thermodynamics and statistical mechanics. The most widely used definition of statistical mechanical entropy for a quantum system is introduced by von Neumann. While in classical systems, the statistical mechanical entropy is defined by Gibbs. The relation between these two definitions of entropy is still not fully explored. In this work, we study this problem by employing the phase-space formulation of quantum mechanics. For those quantum states having well-defined classical counterparts, we study the quantum-classical correspondence and quantum corrections of the entropy. We expand the von Neumann entropy in powers of ℏ by using the phase-space formulation, and the zeroth-order term reproduces the Gibbs entropy. We also obtain the explicit expression of the quantum corrections of the entropy. Moreover, we find that for the thermodynamic equilibrium state, all terms odd in ℏ are exactly zero. As an application, we derive quantum corrections for the net work extraction during a quantum Carnot cycle. Our results bring important insights into the understanding of quantum entropy and may have potential applications in the study of quantum heat engines.
Quantum thermodynamics has emerged as a central field for understanding how energy conversion processes occur in microscopic systems. In these systems, effects such as coherence, entanglement, and non-Markovianity play key roles. In this thesis, we explore different ways to describe quantum thermodynamics, using two main approaches: one based on entropy and the other on ergotropy. First, we introduce a generalized approach to quantify non-Markovianity through the breakdown of monotonicity in thermodynamic functions. In this context, the entropy-based heat flow serves as a practical tool to witness and measure quantum memory in unital maps that do not reverse the sign of the internal energy. Next, we analyze the dynamics of ergotropy in open qubits under both Markovian and non-Markovian evolutions. We identify phenomena such as freezing and sudden death of ergotropy, and we establish an analytical relation between the change in ergotropy and the environment-induced work . This provides a clear physical interpretation for the additional term in the first law in the entropy-based formulation. Finally, we propose an ergotropy-based thermodynamic formulation, in which heat is reinterpreted in terms of the change of the passive state associated with the density operator governing the quantum dynamics. This approach allows one to measure non-Markovianity of unital maps more generally and accurately than the entropy-based heat flow. This advantage comes from the direct link between heat and von Neumann entropy, a property ensured by invariance under passive transformations. Moreover, the out-of-equilibrium temperature naturally remains non-negative, similarly to equilibrium thermodynamics.
This paper introduces a generalized definition of quantum information entropy, aiming to provide a more comprehensive framework for quantifying information in complex quantum systems. The core of this work lies in extending the conventional concept of entropy, traditionally applied to classical information, to the realm of quantum mechanics. We explore various types of quantum entropy, including von Neumann entropy, relative entropy, and a newly proposed generalized entropy, denoted as *Eg*. The definition of *Eg* is based on a novel approach that considers the entanglement structure and correlations within quantum states. We establish relationships between these different entropy measures, demonstrating their interconnectedness within the broader context of quantum information theory. The paper further outlines a mathematical formalism for calculating *Eg*, providing a tool for analyzing and characterizing quantum information. Our generalized entropy provides a more flexible and potentially more accurate measure of quantum information, particularly in scenarios involving complex quantum correlations and multi-partite quantum systems. ---
A central open question in quantum thermodynamics is the identification of thermodynamic entropy away from equilibrium. Although thermodynamic entropy coincides with the von Neumann entropy for a Gibbs state at equilibrium, the thermodynamic role of quantum coherence generated during nonequilibrium evolution remains unclear. Here, we show that von Neumann entropy does not, in general, represent the thermodynamic entropy governing heat exchange in coherent nonequilibrium quantum systems. By integrating the resource theory of quantum coherence with thermodynamic principles, we develop a general framework for nonequilibrium quantum thermodynamics. We identify thermodynamic entropy with the energy entropy derived from energy-measurement statistics and introduce a dynamical temperature extending the equilibrium concept to nonequilibrium states. Consequently, we obtain an exact entropy-balance relation separating the entropy flux associated with heat exchange from the coherence-loss contribution to entropy production, establishing an operational connection between quantum coherence, heat flow, and total entropy change. Unlike approaches relying on equilibrium reference states or stationary Gibbs ensembles, our framework uses instantaneous dynamical quantities and remains valid throughout transient nonequilibrium evolution. It provides consistent dynamical definitions of temperature, work, heat, and free energy. Applying the formalism to an exactly solvable open quantum system, we show that the first and second laws hold throughout transient evolution and that equilibrium thermodynamics emerges dynamically in the weak-coupling long-time limit. Our results establish a unified framework for coherent nonequilibrium quantum thermodynamics and suggest opportunities for controlling heat dissipation through coherence engineering in quantum thermal technologies.