The resource-theoretic approach to quantum thermodynamics assumes complete knowledge of the thermal equilibrium against which thermodynamic resources are defined. In practice, however, this state is determined by the system Hamiltonian and the bath temperature, neither of which is known with perfect precision. We develop a framework in which the equilibrium reference is specified by a set of candidate states reflecting this uncertainty. Under a generic geometric condition, we prove a no-go theorem that sharply limits athermality ``purification'': conversion from an uncertain athermality resource to a definite target is either trivial or impossible, with no room for tradeoff. We then introduce two complementary battery models: a clean battery with a precisely known equilibrium state and a dirty battery with an uncertain one. For both models, we derive exact one-shot entropic characterizations of work extraction and work of formation in terms of standard min- and max-relative entropies and new subspace-constrained variants. In the asymptotic regime, both models exhibit a strong form of thermodynamic irreversibility. In particular, we give a simple and explicit example in which, in the clean-battery model, work is required to form a state but no work can be extracted from it, in direct analogy with bound entanglement, whereas in the dirty-battery model, work can be extracted but formation requires infinite work cost. These phenomena persist even under arbitrarily small uncertainty, showing that equilibrium uncertainty is not a minor perturbation of the standard theory but a qualitatively new ingredient that reshapes the fundamental limits of thermodynamic resource interconversion.
The resource-theoretic approach to quantum thermodynamics assumes complete knowledge of the thermal equilibrium against which thermodynamic resources are defined. In practice, however, this state is determined by the system Hamiltonian and the bath temperature, neither of which is known with perfect precision. We develop a framework in which the equilibrium reference is specified by a set of candidate states reflecting this uncertainty. Under a generic geometric condition, we prove a no-go theorem that sharply limits athermality ``purification'': conversion from an uncertain athermality resource to a definite target is either trivial or impossible, with no room for tradeoff. We then introduce two complementary battery models: a clean battery with a precisely known equilibrium state and a dirty battery with an uncertain one. For both models, we derive exact one-shot entropic characterizations of work extraction and work of formation in terms of standard min- and max-relative entropies and new subspace-constrained variants. In the asymptotic regime, both models exhibit a strong form of thermodynamic irreversibility. In particular, we give a simple and explicit example in which, in the clean-battery model, work is required to form a state but no work can be extracted from it, in direct analogy with bound entanglement, whereas in the dirty-battery model, work can be extracted but formation requires infinite work cost. These phenomena persist even under arbitrarily small uncertainty, showing that equilibrium uncertainty is not a minor perturbation of the standard theory but a qualitatively new ingredient that reshapes the fundamental limits of thermodynamic resource interconversion.
We introduce the resource-theoretic free energy of a quantum channel as the maximal work extractable from the channel as its output equilibrates to a thermal state and its reference system remains locally intact. It is proportional to the relative entropy between the given channel and the absolutely thermal channel. It attains a clear operational meaning as twice the asymptotic rates of athermality distillation and formation under Gibbs-preserving superchannels, which map one absolutely thermal channel to another for a given bath, thereby revealing the asymptotic reversibility of the resource theory of athermality for quantum channels. Consequently, we establish that the optimal extractable work in converting one channel to another through the asymptotic athermality distillation and formation tasks equals the difference in their free energies. We call this optimal work the thermodynamic work capacity of channel conversion. Quantum information processing and computing fundamentally concern the manipulation and transformation of quantum channels, which encompass quantum states, their transformations, and measurements. A quantitative characterization of the optimal thermodynamic work gain or expenditure in quantum information processing constitutes a key step toward formulating thermodynamics of quantum processes.
Within quantum information, a very productive way to look at physical phenomena is under the light of the framework of resource theories. Through it, one can focus on questions such as in what sense a particular physical feature can be thought of as a resource for certain tasks, and how can it be quantified. A particularly prominent example of this is the theory of quantum entanglement. In recent years, a research program has emerged in which researchers are trying to build a resource theory relevant for thermodynamical and non-equilibrium phenomena at the quantum and nano scales. The overarching aim is to find how the perspective given by quantum information tools and techniques help us understand new and existing results of the field. In this thesis we advance these efforts by introducing different kinds of non-equilibrium fluctuations into the theory: fluctuations of work and fluctuations of states. In this way, we manage to merge a number of ideas from the field of stochastic thermodynamics into the resource-theoretic framework. We further explore the structure of this resource theory by introducing results in which the finiteness of the heat bath's size is considered, deriving corrections to well-known expressions regarding work extraction. On top of this, we also consider the usefulness within non-equilibrium thermodynamics of other tools coming from quantum information theory, such as the relative entropy and the idea of a recovery map. The connection between the recovery map and the property of quantum detailed balance (and time reversal more generally) is highlighted, and a bound on the entropy production of Davies maps is derived inspired by recent results on quantum information.
Enhancing the precision of a thermodynamic process inevitably necessitates a thermodynamic cost. This notion was recently formulated as the thermodynamic uncertainty relation, which states that the lower bound on the relative variance of thermodynamic currents decreases as entropy production increases. From another viewpoint, the thermodynamic uncertainty relation implies that if entropy production were allowed to become infinitely large, the lower bound on the relative variance could approach zero. However, it is evident that realizing infinitely large entropy production is infeasible in reality. This indicates that physical constraints impose precision limits on the system, independent of its dynamics. In this study, we derive fundamental precision limits, dynamics-independent bounds on the relative variance and the expectations of observables, for open quantum thermal machines operating within a finite-dimensional system and environment. These bounds are set by quantities such as dimensions, energy bandwidth, and initial minimum eigenvalue, which depend only on the initial configuration and are independent of the dynamics. Using a quantum battery model, the fundamental precision limits show that there is a trade-off between the amount of energy storage and the charging precision. Additionally, we investigate how quantum coherence affects these fundamental limits, demonstrating that the presence of coherence can improve the precision limits. Our findings provide insights into fundamental limits on the precision of quantum thermal machines.
Chapter 17 delves into quantum thermodynamics, building on the concepts introduced in the resource theory of nonuniformity. This chapter focuses on thermal states and athermality as resources within the quantum domain, emphasizing the significance of Gibbs states and their role in quantum statistical mechanics. It outlines the operational framework for thermal operations, setting the stage for discussions on energy conservation and the second law of thermodynamics in quantum systems. A key aspect of the chapter is the exploration of quasi-classical athermality, illustrating how quantum states deviate from thermal equilibrium when the state of the system commutes with its Hamiltonian. In the fully quantum domain, the chapter introduces closed formulas for quantifying athermality, such as the athermality cost and distillable athermality, both in the single-shot and the asymptotic domains. These measures provide a quantitative understanding of the efficiency of thermal operations and the potential for work extraction or consumption.
Uncertainty relations represent a foundational principle in quantum mechanics, imposing inherent limits on the precision with which \textit{mechanically} conjugate variables such as position and momentum can be simultaneously determined. This work establishes analogous relations for \textit{thermodynamically} conjugate variables -- specifically, a classical intensive parameter θ and its corresponding extensive quantum operator O^ -- in equilibrium states. We develop a framework to derive a rigorous thermodynamic uncertainty relation for such pairs, where the uncertainty of the classical parameter θ is quantified by its quantum Fisher information Fθ. The framework is based on an exact integral representation that relates Fθ to the autocorrelation function of operator O^. From this representation, we derive a tight upper bound for the quantum Fisher information, which yields a thermodynamic uncertainty relation: ΔθΔO≥kBT with ΔO≡∂θ⟨O^⟩Δθ and T is the system temperature. The result establishes a fundamental precision limit for quantum sensing and metrology in thermal systems, directly connecting it to the thermodynamic properties of linear response and fluctuations.
The principle of superposition is one of the main building blocks of quantum physics and has tremendous consequences both for our fundamental understanding of nature and for technological applications. In particular, the existence of coherent superposition leads to the concept of unavoidable quantum uncertainty. The role played by this "coherent" uncertainty within thermodynamics, as well as its relationship to classical lack of knowledge, is the main subject of this thesis. In Part I we study thermodynamic limitations of processing quantum coherence within a resource-theoretic framework. Using the time-translation symmetry that arises from the first law of thermodynamics, we find constraints on possible manipulations of coherence and prove their irreversibility due to the second law. We also generalise to the quantum domain Szilard's concept of converting information into work. Namely, we show how, in the presence of a heat bath, coherence of a system can be exploited to perform mechanical work. Finally, we analyse the effect that coherence has on the structure of the thermodynamic arrow of time, i.e., on the set of states into which a given state can freely evolve under thermodynamic constraints. In Part II we focus on the interplay between quantum and classical uncertainty manifested in uncertainty relations. We show that separating the total uncertainty into these two distinct components leads to a new type of "fixed-entropy" uncertainty relation. We also analyse how classical ignorance affects the structure of states that minimise the unavoidable uncertainty arising from the noncommutativity of two observables. Finally, we study error-disturbance trade-off relations and, by proving that quantum uncertainty can be simultaneously maximised for any two observables, we clarify the unphysical nature of state-dependent relations.