Abstract Thermodynamics was developed when it was still unclear whether the microworld had an atomic substructure. The objective here is to investigate the relationship between the process of a quantum measurement and the second law of thermodynamics. It is necessary to make the ideas of heat and work explicit in this approach. It is possible to study the correspondence between information entropy and thermodynamic entropy. Finally, a model is proposed in such a way as to resemble an experimental example for the discussion.
Abstract Thermodynamics was developed when it was still unclear whether the microworld had an atomic substructure. The objective here is to investigate the relationship between the process of a quantum measurement and the second law of thermodynamics. It is necessary to make the ideas of heat and work explicit in this approach. It is possible to study the correspondence between information entropy and thermodynamic entropy. Finally, a model is proposed in such a way as to resemble an experimental example for the discussion.
To understand in detail the relation between unitary quantum theory that describes our world at the microscopic scale and thermodynamics, which was long believed only to apply to macroscopic objects, is one of the most interesting and long-standing problems in physics. Recently, this problem has received renewed attention, in particular from the community of quantum information theory, but also from the field of statistical mechanics, inspired from stochastic thermodynamics. These results suggest that thermodynamics is also relevant for individual quantum systems, provided that they can be brought into contact with thermal baths. In this thesis, I use recently developed tools to provide new results both on fundamental questions, but also on questions which are of practical relevance for potential miniaturized thermal machines. In terms of fundamental questions, the results in this thesis contribute to understanding how the basic laws of thermodynamics and statistical mechanics can be understood directly from unitary quantum mechanics. In particular, I discuss and answer the following questions: i) How can we quantify the third law of thermodynamics using information theoretic methods? ii) How can we quantify the "thermodynamic value" of a state-transition in quantum systems? iii) How can we axiomatically characterize the non-equilibrium free energy and relative entropy? iv) How can we justify statistical ensembles from an operational perspective, without having to introduce either some probability measures or an information theoretic entropy measure? v) How can we understand the equilibration of closed quantum systems, how long does it take and how difficult is it to avoid? In the second part of the thesis I discuss in detail how experimental restrictions, which become important at the quantum scale, influence the ultimate thermodynamic bounds for thermal machines. In particular, the results in this thesis provide thermodynamic bounds on work-extraction and efficiencies of thermal machines in situations where 1.) an experimenter only has limited field strengths available, 2.) an experimenter cannot control the interactions between particles, but external fields arbitrarily well, and 3.) situations in which a small quantum system can only be strongly coupled to heat baths. These bounds are tight and I provide explicit examples illustrating the different behaviours. Finally, I come back to a classic problem in statistical physics: The emergence of spontaneous symmetry breaking. Here, I provide general and rigorous new results that show how symmetry-breaking stationary states emerge from fluctuations in order parameters in dissipative lattice models.
We now turn to the subject of thermodynamics, and to describing the process of thermalization of quantum systems. A quantum system that interacts with another system large enough to obey the usual rules of thermodynamics, is referred to as an open system. The thermal system with which it interacts is often referred to as a thermal bath , or simply as the environment. We will use these terms interchangeably. Modeling the behavior of open quantum systems has many applications. One of these is to model the noise that they are subjected to by their surroundings, noise that we will wish to control using measurement and feedback loops in Chapter 5. There are also two important connections between quantum measurements, information theory, and thermodynamics. The first of these is that the thermodynamic entropy of a macroscopic system can be identified with the information-theoretic entropy that characterizes the lack of knowledge of an observer about the microstates of the system. In turn, if an observer is allowed to make microscopic measurements on the system, then this modifies the thermodynamic rules regarding the conversion of heat to work. Such is the subject of Maxwell’s demon. Curiously it took more than a century to understand how Maxwell’s demon, essentially a microscopic measurement and feedback process, fits within the second law of thermodynamics. In another twist, it turns out that the process of erasing information requires an increase in the thermodynamic entropy of the environment, and this increase is directly related to the amount of information erased.
In Chapter 3 the introduction of entropy as a state function was facilitated by the realization that there exist possible and impossible processes, and by an examination of the heat and work effects occurring during these processes. From the formal statement of the Second Law of Thermodynamics, as developed from Classical Thermodynamics arguments, it is difficult to assign a physical significance or a physical quality to entropy. In this respect entropy differs from internal energy in spite of the fact that, within the scope of Classical Thermodynamics, both properties are simply mathematical functions of the state of a system. The ready acceptance of the First Law of Thermodynamics, after its enunciation, was due to the easily understood physical significance of internal energy, whereas the lack of corresponding understanding of entropy caused the acceptance of the Second Law of Thermodynamics to be slow. From the classical viewpoint the Second Law is only valid as a direct result of the fact that, to date, human ingenuity has failed to invent a perpetual motion machine. Thus, within the scope of Classical Thermodynamics, the Second Law is a “law” only because it has not yet been disproved. The physical interpretation of entropy had to await the development of quantum theory and statistical mechanics.
We investigate fundamental connections between thermodynamics and quantum information theory. First, we show that the operational framework of thermal operations is nonequivalent to the framework of Gibbs-preserving maps, and we comment on this gap. We then introduce a fully information-theoretic framework generalizing the above by making further abstraction of physical quantities such as energy. It is technically convenient to work with and reproduces known results for finite-size quantum thermodynamics. With our framework we may determine the minimal work cost of implementing any logical process. In the case of information processing on memory registers with a degenerate Hamiltonian, the answer is given by the max-entropy, a measure of information known from quantum information theory. In the general case, we obtain a new information measure, the "coherent relative entropy", which generalizes both the conditional entropy and the relative entropy. It satisfies a collection of properties which justifies its interpretation as an entropy measure and which connects it to known quantities. We then present how, from our framework, macroscopic thermodynamics emerges by typicality, after singling out an appropriate class of thermodynamic states possessing some suitable reversibility property. A natural thermodynamic potential emerges, dictating possible state transformations, and whose differential describes the physics of the system. The textbook thermodynamics of a gas is recovered as well as the form of the second law relating thermodynamic entropy and heat exchange. Finally, noting that quantum states are relative to the observer, we see that the procedure above gives rise to a natural form of coarse-graining in quantum mechanics: Each observer can consistently apply the formalism of quantum information according to their own fundamental unit of information.