The central purpose of this thesis is to study the familiar processes of quantum information theory from a thermodynamic point of view. Chapter 1 introduces equilibrium thermodynamics and its laws. The laws are then motivated in a statistical mechanics approach which explicitly employs the language of quantum information theory. Chapter 2 is dedicated to the concept of work extraction, in particular in non- equilibrium scenarios. In this context a generalised notion of equilibrium called passivity is introduced. The chapter’s final section gives an introduction and an overview of various approaches to work extraction ranging from classical to explicitly quantum scenarios. Chapter 3 considers the question of powerful unitary operations. Focussing on the case of qubits (or rather wits, short for ‘work bits’) the chapter’s first result is an explicit protocol for maximally powerful driving for any given initial state. Building onto this, the chapter’s second main result is a proof that the evolution time (and hence the power of the corresponding work deposition process) can be decreased by a factor of 1/N for an array of N wits if global driving is permitted (rather than individual driving in parallel). Lastly, the thermodynamics of completely-positive and trace-preserving (CPTP) maps is considered in Chapter 4. It is shown that the energy change during such a process can be split into three parts: a work-like equilibrium contribution, a heat-like equilibrium contribution, and a third genuinely non-equilibrium energy difference. All three terms obtain their meaning when considering how much work can be extracted from the input and output state of the CPTP process, respectively. Furthermore, the identification of work and heat in this manner complies with a recently published version of the second law, which applies to the same class of processes. This is demonstrated by considering unital and thermal maps.
The central purpose of this thesis is to study the familiar processes of quantum information theory from a thermodynamic point of view. Chapter 1 introduces equilibrium thermodynamics and its laws. The laws are then motivated in a statistical mechanics approach which explicitly employs the language of quantum information theory. Chapter 2 is dedicated to the concept of work extraction, in particular in non- equilibrium scenarios. In this context a generalised notion of equilibrium called passivity is introduced. The chapter’s final section gives an introduction and an overview of various approaches to work extraction ranging from classical to explicitly quantum scenarios. Chapter 3 considers the question of powerful unitary operations. Focussing on the case of qubits (or rather wits, short for ‘work bits’) the chapter’s first result is an explicit protocol for maximally powerful driving for any given initial state. Building onto this, the chapter’s second main result is a proof that the evolution time (and hence the power of the corresponding work deposition process) can be decreased by a factor of 1/N for an array of N wits if global driving is permitted (rather than individual driving in parallel). Lastly, the thermodynamics of completely-positive and trace-preserving (CPTP) maps is considered in Chapter 4. It is shown that the energy change during such a process can be split into three parts: a work-like equilibrium contribution, a heat-like equilibrium contribution, and a third genuinely non-equilibrium energy difference. All three terms obtain their meaning when considering how much work can be extracted from the input and output state of the CPTP process, respectively. Furthermore, the identification of work and heat in this manner complies with a recently published version of the second law, which applies to the same class of processes. This is demonstrated by considering unital and thermal maps.
Accurately describing work extraction from a quantum system is a central objective for the extension of thermodynamics to individual quantum systems. The concepts of work and heat are surprisingly subtle when generalizations are made to arbitrary quantum states. We formulate an operational thermodynamics suitable for application to an open quantum systems undergoing a general quantum evolution. We derive the first law of ther- modynamics for a process described by a completely-positive and trace-preserving map and show consistency with the Hatano-Sasa statement of the second law. We show that heat, from the first law, is positive when the input state of the map majorises the output state. Moreover, the change in entropy is also positive for the same majorisation condition. This makes a strong connection between the two operational laws of thermodynamics. Introduction.— The laws of thermodynamics were forged in the furnaces of the industrial revolution, as engineers and scientists refined their picture of energy, studying heat and its interconversion to mechanical work with a view to powering the mines and factories of this new era of human endeavor. Followed by the development of statistical mechanics at the change of the centuries (1), far from its pragmatic inception, thermodynamics is now a theory with a remarkable range of applicability, successfully describing the properties of macro- scopic systems ranging from refrigerators to black holes (2). Moving on to the 21st century with both the industrial and electronic revolutions behind us, we are currently pushing technology towards and beyond the microscopic scale. With a view to devices operating at a scale where quantum mech- anical laws become important we may ask whether the solid and some combination of the two (8). Finally, central to the work presented here is a work extraction formalism for non-passivity of quantum states (9). Despite the range of ap- proaches a more general picture for the thermodynamics of general quantum evolutions is far from clear. In this Letter, we take an operational approach to character- izing the energy change of an open quantum process described by a completely-positive trace-preserving (CPTP) map. In analogy to the first law of thermodynamics we discuss work done, extractable work, and heat. The concepts of ergotropy and adiabatic work allow us to state our main result: An op- erational first law for general quantum processes. We show that our operational first law is in agreement with widely used Hatano-Sasa version of the second law for CPTP maps (10, 11) by explicitly stating the Clausius inequality for unital and thermal maps. We then show that both operational heat and the change in von Neumann entropy are positive when the input state of the map majorises the output state. Thermodynamics of quantum systems.— The first law of thermodynamics states that the internal energy change in a thermodynamic process can be split up into two contributions - work and heat: dE = Q + W. For a generic quantum system, the internal energy at time t is E(t) = tr( (t)H(t)), implying that the change in the internal energy dE depends only on the end points. Heat and work on the other hand are path-dependent—thus the different notation for the 'differen- tials'. As an illustration we may consider the heat expended when pushing a piston into a cylinder filled with gas: It not only depends on the initial and final positions of the piston but also on how fast it is pushed. Using the derivative of the internal energy with respect to time the following two expres- sions are motivated (10):
Accurately describing work extraction from a quantum system is a central objective for the extension of thermodynamics to individual quantum systems. The concepts of work and heat are surprisingly subtle when generalizations are made to arbitrary quantum states. We formulate an operational thermodynamics suitable for application to an open quantum system undergoing quantum evolution under a general quantum process by which we mean a completely positive and trace-preserving map. We derive an operational first law of thermodynamics for such processes and show consistency with the second law. We show that heat, from the first law, is positive when the input state of the map majorizes the output state. Moreover, the change in entropy is also positive for the same majorization condition. This makes a strong connection between the two operational laws of thermodynamics.
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.
A key concept in quantum thermodynamics is extractable work, which specifies the maximum amount of work that can be extracted from a quantum system. Different quantities are used to measure extractable work, the most prevalent of which are ergotropy and the difference between the non-equilibrium and equilibrium quantum free energies. Using the latter, we investigate the evolution of extractable work when an open quantum system undergoes a general quantum process described by a completely-positive and trace-preserving dynamical map. We derive a fundamental equation of thermodynamics for such processes as a relation between the distinct sorts of energy change in such a way that the first and the second law of thermodynamics are combined. We then identify the contributions from the reversible and irreversible processes in this equation and demonstrate that they are respectively responsible for the evolution of heat and extractable work of the open quantum system. Furthermore, we show how this correspondence between irreversibility and extractable work has the potential to provide a clear explanation of how the quantum properties of a system affect its extractable work evolution. Specifically, we establish this by directly connecting the change in extractable work with the change in standard quantifiers of quantum non-Markovianity and quantum coherence during a general quantum process. We illustrate these results with two examples.
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.
Thermodynamics is a phenomenological theory describing the energy conversion of work and heat. At its origins thermodynamics was developed in order to understand and improve heat engines. In conventional thermodynamics, however, only such processes are completely describable which are slow enough to keep the system of interest in an equilibrium state with its thermal environment at all times. On the contrary, all real physical processes are accompanied by non-equilibrium phenomena. These are mathematically described with the help of the irreversible entropy production, i.e. the dissipated work divided by the temperature of the bath. The second law of thermodynamics merely states that the irreversible entropy production is on average always positive. The modern trend of miniaturization, however, leads to smaller and smaller devices. On short length scales thermal noise as well as quantum fluctuations become important. Thus, usual thermodynamic quantities as work and heat acquire stochastic nature. Moreover, in the quantum regime a completely new theory had to be invented, since classical notions of work and heat are no longer valid. The present dissertation contributes to this prevailing field by the derivation of analytical expressions for the entropy production in open and closed quantum system far from thermal equilibrium. The theoretical treatment is motivated by an experimental point of view. The obtained results were derived from the reduced dynamics of the quantum mechanical system, which may be coupled to a thermal bath. To this end, it was dealt with methods and approaches of statistical physics, conventional thermodynamics, quantum information theory and the theory of open quantum systems. In this dissertation the most important physical systems are treated. First, an isolated system is considered before later on a heat bath is included. The limit of weak as well as of strong coupling are analyzed in detail. For isolated systems it turns out that a geometric approach is able to capture the thermodynamic properties, completely. To this end, the irreversible entropy production is sharply estimated from below by the angle between the real non-equilibrium state and the corresponding equilibrium one. Further, it is shown that the Heisenberg uncertainty relation for energy and time can be formulated more precisely in terms of that angle. As experimental system cold ion traps are introduced, in which the theoretical predictions can be verified. For weakly coupled systems, then, the irreversible entropy production is identified and a fluctuation theorem is derived by merely thermodynamic arguments. As experimental system we, again, propose the cold ion traps, where the heat bath can be simulated with the help of LASER light. In the limit of strong coupling a semiclassical description of the reduced quantum system is discussed. Again, an analytically exact expression for the entropy production and the corresponding fluctuation theorem is derived. A physical system, in which strong coupling to the environment can be analyzed, is given by Josephson junctions. This dissertation discusses the entropy production for isolated, weakly coupled, and overdamped quantum systems. For these physical regimes analytical relation are derived and experimental systems proposed, with which the predictions can be verified.