The resource-theoretic approach to quantum thermodynamics typically assumes perfect knowledge of the thermal equilibrium state, an idealization incompatible with finite experimental precision. We develop a framework for equilibrium uncertainty by representing the equilibrium reference as a set of candidate states. Under a generic geometric condition, we prove a no-go theorem that sharply limits athermality ``purification'': converting an uncertain athermal state into a definite target is either trivial or impossible. We then derive exact one-shot entropic characterizations of work extraction and formation for two work-storage models, a clean battery with known equilibrium and a dirty battery with uncertain equilibrium. Both models exhibit strong asymptotic irreversibility even under arbitrarily small uncertainty. An explicit example reveals two distinct extremes: clean batteries display a bound-entanglement-like phenomenon, with positive formation cost but zero extractable work, whereas dirty batteries allow positive work extraction but require infinite formation cost. These phenomena show that equilibrium uncertainty is not a minor perturbation of the standard theory, but a structural ingredient that fundamentally reshapes the limits of quantum thermodynamics.
The resource-theoretic approach to quantum thermodynamics typically assumes perfect knowledge of the thermal equilibrium state, an idealization incompatible with finite experimental precision. We develop a framework for equilibrium uncertainty by representing the equilibrium reference as a set of candidate states. Under a generic geometric condition, we prove a no-go theorem that sharply limits athermality ``purification'': converting an uncertain athermal state into a definite target is either trivial or impossible. We then derive exact one-shot entropic characterizations of work extraction and formation for two work-storage models, a clean battery with known equilibrium and a dirty battery with uncertain equilibrium. Both models exhibit strong asymptotic irreversibility even under arbitrarily small uncertainty. An explicit example reveals two distinct extremes: clean batteries display a bound-entanglement-like phenomenon, with positive formation cost but zero extractable work, whereas dirty batteries allow positive work extraction but require infinite formation cost. These phenomena show that equilibrium uncertainty is not a minor perturbation of the standard theory, but a structural ingredient that fundamentally reshapes the limits of quantum thermodynamics.
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.
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.
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.
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.
The manipulation of quantum "resources" such as entanglement, coherence, and magic states lies at the heart of quantum science and technology, empowering potential advantages over classical methods. In practice, a particularly important kind of manipulation is to "purify" the quantum resources since they are inevitably contaminated by noise and thus often lose their power or become unreliable for direct usage. Here we prove fundamental limitations on how effectively generic noisy resources can be purified enforced by the laws of quantum mechanics, which universally apply to any reasonable kind of quantum resource. More explicitly, we derive nontrivial lower bounds on the error of converting any full-rank noisy state to any target pure resource state by any free protocol (including probabilistic ones)-it is impossible to achieve perfect resource purification, even probabilistically. Our theorems indicate strong limits on the efficiency of distillation, a widely used type of resource purification routine that underpins many key applications of quantum information science. In particular, this general result induces the first explicit lower bounds on the resource cost of magic state distillation, a leading scheme for realizing scalable fault-tolerant quantum computation. Implications for the standard error-correction-based methods are specifically discussed.
Thermal operations are quantum channels that play a central role in deriving thermodynamic limitations in quantum systems. However, they were originally defined by implementation procedures rather than by fundamental principles. Alternative models of thermal processes have been proposed, but they obscure the resources required for implementation. Here, we identify the universal principle that separates these models. We show that thermal operations are uniquely characterized by a purely quantum information-theoretic property, which we call informational equilibrium. This formulation does not assume a particular Hamiltonian or even a notion of temperature: a channel is a thermal operation precisely when it leaves its environment invariant whenever the system is prepared in equilibrium. In this way, the usual energetic description of equilibrium (i.e., Gibbs states) emerges as a representation of an underlying informational constraint. Extending this framework, we note that enforcing environment invariance for all system inputs leads to catalytic channels, which idealize perfectly reusable heat baths. This informational equilibrium perspective culminates in a hierarchy for doubly-stochastic quantum channels, with strict separations between the classes. Our refined hierarchy sharpens the failure of the Birkhoff-von Neumann theorem in the quantum regime, and highlights the richer structural landscape of quantum channels compared to classical stochastic processes that cannot be captured by state-convertibility relations.