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 then explores typical thermodynamic settings, such as cycles and work extraction protocols, when the working material is genuinely quantum. Finally, Chapter 3 explores the thermodynamics of quantum information processing and introduces the reader to some more state-of-the-art topics in this exciting and rapidly developing research field.
After a brief historical introduction of quantum thermodynamics, we present its main goal and objectives, from fundamental questions to practical applications in quantum technologies. We then introduce the standard thermodynamic framework for quantum systems, valid in the weak coupling limit and in thermal environments. Extensions to strong coupling and nonthermal baths are briefly mentioned. Then, using this standard thermodynamic framework, we describe the quantum version of the Otto cycle and Carnot cycle for heat-to-work conversion as well as for refrigeration. The analysis of their differences is used to introduce the concept of entropy production at the quantum level, which quantifies irreversibility and is responsible for loss of efficiency. Then, we present a simple description of an autonomous quantum refrigerator, a promising device for qubit re-initialisation in quantum technologies. Finally, we briefly mention some quantum effects in heat engines and refrigerators. We conclude with a perspective on the main goals of quantum thermodynamics.
The ground state of a two-level system (associated with probabilities
p and $1\ensuremath{-}p$, respectively) defined by a general Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}={\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{0}+\ensuremath{\lambda}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}$ is studied. The simple case characterized by $\ensuremath{\lambda}=0$, whose Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{0}$ is represented by a diagonal matrix, is well established and solvable within Boltzmann-Gibbs statistical mechanics; in particular, it follows the third law of thermodynamics, presenting zero entropy (
SBG=0) at zero temperature (
T=0). Herein it is shown that the introduction of a perturbation $\ensuremath{\lambda}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}$ ($\ensuremath{\lambda}>0$) in the Hamiltonian may lead to a nontrivial ground state, characterized by an entropy
S[p] (with $S[p]\ensuremath{\ne}{S}_{\mathrm{BG}}[p]$), if the Hermitian operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}$ is represented by a $2\ifmmode\times\else\texttimes\fi{}2$ matrix, defined by nonzero off-diagonal elements ${V}_{12}={V}_{21}=\ensuremath{-}z$, where
z is a real positive number. Hence, this new term in the Hamiltonian, presenting ${V}_{12}\ensuremath{\ne}0$, may produce physically significant changes in the ground state, and especially, it allows for the introduction of an effective temperature $\ensuremath{\theta}$ ($\ensuremath{\theta}\ensuremath{\propto}\ensuremath{\lambda}z$), which is shown to be a parameter conjugated to the entropy
S. Based on this, one introduces an infinitesimal heatlike quantity, $\ensuremath{\delta}Q=\ensuremath{\theta}dS$, leading to a consistent thermodynamic framework, and by proposing an infinitesimal form for the first law, a Carnot cycle and thermodynamic potentials are obtained. All results found are very similar to those of usual thermodynamics, through the identification $T\ensuremath{\leftrightarrow}\ensuremath{\theta}$, and particularly the form for the efficiency of the proposed Carnot Cycle. Moreover,
S also follows a behavior typical of a third law, i.e., $S\ensuremath{\rightarrow}0$, when $\ensuremath{\theta}\ensuremath{\rightarrow}0$.