Asymptotic Reversibility of Thermal Operations for Interacting Quantum Spin Systems via Generalized Quantum Stein's Lemma

For quantum spin systems in any spatial dimension with a local, translation-invariant Hamiltonian, we prove that asymptotic state convertibility from a quantum state to another one by a thermodynamically feasible class of quantum dynamics, called thermal operations, is completely characterized by the Kullback-Leibler (KL) divergence rate, if the state is translation-invariant and spatially ergodic. Our proof consists of two parts and is phrased in terms of a branch of the quantum information theory called the resource theory. First, we prove that any states, for which the min and max Rényi divergences collapse approximately to a single value, can be approximately reversibly converted into one another by thermal operations with the aid of a small source of quantum coherence. Second, we prove that these divergences collapse asymptotically to the KL divergence rate for any translation-invariant ergodic state. We show this via a generalization of the quantum Stein's lemma for quantum hypothesis testing beyond independent and identically distributed (i.i.d.) situations. Our result implies that the KL divergence rate serves as a thermodynamic potential that provides a complete characterization of thermodynamic convertibility of ergodic states of quantum many-body systems in the thermodynamic limit, including out-of-equilibrium and fully quantum situations

Asymptotic reversibility of thermal operations for interacting quantum spin systems via generalized quantum Stein’s lemma

For quantum spin systems in any spatial dimension with a local, translation-invariant Hamiltonian, we prove that asymptotic state convertibility from a quantum state to another one by a thermodynamically feasible class of quantum dynamics, called thermal operations, is completely characterized by the Kullback–Leibler (KL) divergence rate, if the state is translation-invariant and spatially ergodic. Our proof consists of two parts and is phrased in terms of a branch of the quantum information theory called the resource theory. First, we prove that any states, for which the min and max Rényi divergences collapse approximately to a single value, can be approximately reversibly converted into one another by thermal operations with the aid of a small source of quantum coherence. Second, we prove that these divergences collapse asymptotically to the KL divergence rate for any translation-invariant ergodic state. We show this via a generalization of the quantum Stein's lemma for quantum hypothesis testing beyond independent and identically distributed situations. Our result implies that the KL divergence rate serves as a thermodynamic potential that provides a complete characterization of thermodynamic convertibility of ergodic states of quantum many-body systems in the thermodynamic limit, including out-of-equilibrium and fully quantum situations

Catalysis in Quantum Information Theory

Catalysts open up new reaction pathways which can speed up chemical reactions while not consuming the catalyst. A similar phenomenon has been discovered in quantum information science, where physical transformations become possible by utilizing a (quantum) degree of freedom that remains unchanged throughout the process. In this review, we present a comprehensive overview of the concept of catalysis in quantum information science and discuss its applications in various physical contexts.Comment: Review paper; Comments and suggestions welcome

Quantum Coarse-Graining: An Information-Theoretic Approach to Thermodynamics

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.Comment: Ph. D. thesis, ETH Zurich (301 pages). Chaps. 1-3,9 are introductory and/or reviews; Chaps. 4,6 discuss previously published results (reproduces content from arXiv:1406.3618, New J. Phys. 2015 and from arXiv:1211.1037, Nat. Comm. 2015); Chaps. 5,7,8,10 are as of yet unpublished (introducing our information-theoretic framework, the coherent relative entropy, and quantum coarse-graining

From ground state cooling to spontaneous symmetry breaking

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.Eines der spannendsten Probleme in der Physik ist zu verstehen wie genau die Thermodynamik aus der mikroskopischen Quantentheorie hervorgeht. Dieses klassische Problem hat in den letzten Jahren erneute Aufmerksamkeit erfahren, einerseits aus Sicht der Quanteninformationstheorie, andererseits aus Sicht der statistischen Mechanik, insbesondere motiviert durch Ergebnisse der stochastischen Thermodynamik. Die Ergebnisse dieser Arbeiten deuten darauf hin, dass thermodynamische Konzepte nicht nur für makroskopische Systeme, sondern auch für ein einzelne Quantensysteme relevant sind, wenn diese in Kontakt mit Wärmebädern gebracht werden können. In dieser Dissertation verwende ich kürzlich entwickelte Methoden, um sowohl neue Resultate in Bezug auf fundamentale Fragestellungen, als auch Resultate welche für potentielle mikroskopische thermische Maschinen relevant sind, herzuleiten. Die Resultate in Bezug auf fundamentale Fragestellungen helfen dabei zu verstehen wie Thermodynamik und statistische Mechanik aus der unitären Quantenmechanik heraus verstanden werden können. Insbesondere diskutiere (und beantworte ich) dabei die folgenden Fragen: i) Wie können wir den dritten Hauptsatz der Thermodynamik mithilfe von informationstheoretischen Methoden quantifizieren? ii) Wie lässt sich der "thermodynamische Wert" von Zustandsänderungen in Quantensystemen aus operationaler Sichtweise quantifizieren? iii) Wie können wir die freie Energie sowie die relative Entropie für Quantensysteme axiomatisch charakterisieren? iv) Wie können kanonische statistische Gesamtheiten aus operationaler Sichtweise gerechtfertigt werden, ohne Wahrscheinlichkeitsmaße oder informationstheoretische Entropien einzuführen? v) Wie können wir das Äquilibrierungsverhalten geschlossener Quantensysteme verstehen, wie lange dauert es bis ein solches System äquilibriert und wie schwierig ist es ein solches Verhalten zu verhindern? Im zweiten Teil der Arbeit diskutiere ich im Detail welche Auswirkungen zusätzliche experimentelle Einschränkungen auf die theoretischen thermodynamischen Schranken für die Effizienz von thermischen Maschinen im Quantenregime haben. Insbesondere diskutiere ich theoretische Schranken für die Extraktion von Arbeit und den Wirkungsgrad von thermischen Maschinen in Situationen in denen 1.) nur beschränkte Feldstärken in einem Experiment zur Verfügung stehen, 2.) in denen ein_e Experimentator_in in der Lage ist externe Felder zu kontrollieren, aber nicht die Wechselwirkung zwischen einzelnen Spins und 3.) Situationen in denen ein Quantensystem nur durch eine starke Wechselwirkung in Kontakt mit einem Wärmebad gebracht werden kann. Diese neuen Schranken sind strikt und ich illustriere sie mit mehreren Beispielen. Schließlich komme ich zurück zu einem klassischen Problem der statistischen Physik: Das Auftreten von spontaner Symmetriebrechung. Hier präsentiere ich allgemeine und rigorose Resultate, welche zeigen wie spontane Symmetriebrechung aus Fluktuationen in lokalen Ordnungsparametern in dissipativen Gittermodellen hervorgeht

Entropic Continuity Bounds & Eventually Entanglement-Breaking Channels

This thesis combines two parallel research directions: an exploration into the continuity properties of certain entropic quantities, and an investigation into a simple class of physical systems whose time evolution is given by the repeated application of a quantum channel. In the first part of the thesis, we present a general technique for establishing local and uniform continuity bounds for Schur concave functions; that is, for real-valued functions which are decreasing in the majorization pre-order. Continuity bounds provide a quantitative measure of robustness, addressing the following question: If there is some uncertainty or error in the input, how much uncertainty is there in the output? Our technique uses a particular relationship between majorization and the trace distance between quantum states (or total variation distance, in the case of probability distributions). Namely, the majorization pre-order attains a maximum and a minimum over ε-balls in this distance. By tracing the path of the majorization-minimizer as a function of the distance ε, we obtain the path of ``majorization flow’’. An analysis of the derivatives of Schur concave functions along this path immediately yields tight continuity bounds for such functions. In this way, we find a new proof of the Audenaert-Fannes continuity bound for the von Neumann entropy, and the necessary and sufficient conditions for its saturation, in a universal framework which extends to the other functions, including the Rényi and Tsallis entropies. In particular, we prove a novel uniform continuity bound for the α-Rényi entropy with α > 1 with much improved dependence on the dimension of the underlying system and the parameter α compared to previously known bounds. We show that this framework can also be used to provide continuity bounds for other Schur concave functions, such as the number of connected components of a certain random graph model as a function of the underlying probability distribution, and the number of distinct realizations of a random variable in some fixed number of independent trials as a function of the underlying probability mass function. The former has been used in modeling the spread of epidemics, while the latter has been studied in the context of estimating measures of biodiversity from observations; in these contexts, our continuity bounds provide quantitative estimates of robustness to noise or data collection errors. In the second part, we consider repeated interaction systems, in which a system of interest interacts with a sequence of probes, i.e. environmental systems, one at a time. The state of the system after each interaction is related to the state of the system before the interaction by the so-called reduced dynamics, which is described by the action of a quantum channel. When each probe and the way it interacts with the system is identical, the reduced dynamics at each step is identical. In this scenario, under the additional assumption that the reduced dynamics satisfies a faithfulness property, we characterize which repeated interaction systems break any initially-present entanglement between the system and an untouched reference, after finitely many steps. In this case, the reduced dynamics is said to be eventually entanglement-breaking. This investigation helps improve our understanding of which kinds of noisy time evolution destroy entanglement. When the probes and their interactions with the system are slowly-varying (i.e. adiabatic), we analyze the saturation of Landauer's bound, an inequality between the entropy change of the system and the energy change of the probes, in the limit in which the number of steps tends to infinity and both the difference between consecutive probes and the difference between their interactions vanishes. This analysis proceeds at a fine-grained level by means of a two-time measurement protocol, in which the energy of the probes is measured before and after each interaction. The quantities of interest are then studied as random variables on the space of outcomes of the energy measurements of the probes, providing a deeper insight into the interrelations between energy and entropy in this setting.Cantab Capital Institute for the Mathematics of Informatio

On Rejection Sampling in Lyubashevsky's Signature Scheme

International audienc