Correlations in typicality and an affirmative solution to the exact catalytic entropy conjecture

It is well known that if a (finite-dimensional) density matrix ρ has smaller entropy than ρ0, then the tensor product of sufficiently many copies of ρ majorizes a quantum state arbitrarily close to the tensor product of correspondingly many copies of ρ0. In this short note I show that if additionally rank(ρ) ≤ rank(ρ0), then n copies of ρ also majorize a state where all single-body marginals are exactly identical to ρ0 but arbitrary correlations are allowed (for some sufficiently large n). An immediate application of this is an affirmative solution of the exact catalytic entropy conjecture introduced by Boes et al. [PRL 122, 210402 (2019)]: If H(ρ) < H(ρ0) and rank(ρ) ≤ rank(ρ0) there exists a finite dimensional density matrix σ and a unitary U such that Uρ⊗ σU has marginals ρ0 and σ exactly. All the results transfer to the classical setting of probability distributions over finite alphabets with unitaries replaced by permutations

Revisiting the equality conditions of the data processing inequality for the sandwiched R\'enyi divergence

We provide a transparent, simple and unified treatment of recent results on the equality conditions for the data processing inequality (DPI) of the sandwiched quantum R\'enyi divergence, including the statement that equality in the data processing implies recoverability via the Petz recovery map for the full range of $\alpha$ recently proven by Jen\v cov\'a. We also obtain a new set of equality conditions, generalizing a previous result by Leditzky et al.Comment: 11 pages, no figures. Comments are welcome. v2: references update

Statistical ensembles without typicality

Maximum-entropy ensembles are key primitives in statistical mechanics from which thermodynamic properties can be derived. Over the decades, several approaches have been put forward in order to justify from minimal assumptions the use of these ensembles in statistical descriptions. However, there is still no full consensus on the precise reasoning justifying the use of such ensembles. In this work, we provide a new approach to derive maximum-entropy ensembles taking a strictly operational perspective. We investigate the set of possible transitions that a system can undergo together with an environment, when one only has partial information about both the system and its environment. The set of all these allowed transitions encodes thermodynamic laws and limitations on thermodynamic tasks as particular cases. Our main result is that the set of allowed transitions coincides with the one possible if both system and environment were assigned the maximum entropy state compatible with the partial information. This justifies the overwhelming success of such ensembles and provides a derivation without relying on considerations of typicality or information-theoretic measures.Comment: 9+9 pages, 3 figure

Axiomatic Characterization of the Quantum Relative Entropy and Free Energy

Building upon work by Matsumoto, we show that the quantum relative entropy with full-rank second argument is determined by four simple axioms: (i) Continuity in the first argument; (ii) the validity of the data-processing inequality; (iii) additivity under tensor products; and (iv) super-additivity. This observation has immediate implications for quantum thermodynamics, which we discuss. Specifically, we demonstrate that, under reasonable restrictions, the free energy is singled out as a measure of athermality. In particular, we consider an extended class of Gibbs-preserving maps as free operations in a resource-theoretic framework, in which a catalyst is allowed to build up correlations with the system at hand. The free energy is the only extensive and continuous function that is monotonic under such free operations. View Full-Tex

Lieb-Robinson bounds imply locality of interactions

Discrete lattice models are a cornerstone of quantum many-body physics. They arise as effective descriptions of condensed matter systems and lattice-regularized quantum field theories. Lieb-Robinson bounds imply that if the degrees of freedom at each lattice site only interact locally with each other, correlations can only propagate with a finite group velocity through the lattice, similarly to a light cone in relativistic systems. Here we show that Lieb-Robinson bounds are equivalent to the locality of the interactions: a system with k-body interactions fulfills Lieb-Robinson bounds in exponential form if and only if the underlying interactions decay exponentially in space. In particular, our result already follows from the behavior of two-point correlation functions for single-site observables and generalizes to different decay behaviours as well as fermionic lattice models. As a side-result, we thus find that Lieb-Robinson bounds for single-site observables imply Lieb-Robinson bounds for bounded observables with arbitrary support.Comment: 4.5 + 7 pages, 1 figure; v2: Changed title, added references, improved presentatio

Towards Holography via Quantum Source-Channel Codes

While originally motivated by quantum computation, quantum error correction (QEC) is currently providing valuable insights into many-body quantum physics, such as topological phases of matter. Furthermore, mounting evidence originating from holography research (AdS/CFT) indicates that QEC should also be pertinent for conformal field theories. With this motivation in mind, we introduce quantum source-channel codes, which combine features of lossy compression and approximate quantum error correction, both of which are predicted in holography. Through a recent construction for approximate recovery maps, we derive guarantees on its erasure decoding performance from calculations of an entropic quantity called conditional mutual information. As an example, we consider Gibbs states of the transverse field Ising model at criticality and provide evidence that they exhibit nontrivial protection from local erasure. This gives rise to the first concrete interpretation of a bona fide conformal field theory as a quantum error correcting code. We argue that quantum source-channel codes are of independent interest beyond holography

Ticking clocks in quantum theory

We present a derivation of the structure and dynamics of a ticking clock by showing that for finite systems a single natural principle serves to distinguish what we understand as ticking clocks from time-keeping systems in general. As a result we recover the bipartite structure of such a clock: that the information about ticks is a classical degree of freedom. We describe the most general form of the dynamics of such a clock, and discuss the additional simplifications to go from a general ticking clock to models encountered in literature. The resultant framework encompasses various recent research results despite their apparent differences. Finally, we introduce the information theory of ticking clocks, distinguishing their abstract information content and the actually accessible information.Comment: 16 pages + 3 pages appendix, 4 figure

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

Imperfect Thermalizations Allow for Optimal Thermodynamic Processes

Optimal (reversible) processes in thermodynamics can be modelled as step-by-step processes, where the system is successively thermalized with respect to different Hamiltonians by an external thermal bath. However, in practice interactions between system and thermal bath will take finite time, and precise control of their interaction is usually out of reach. Motivated by this observation, we consider finite-time and uncontrolled operations between system and bath, which result in thermalizations that are only partial in each step. We show that optimal processes can still be achieved for any non-trivial partial thermalizations at the price of increasing the number of operations, and characterise the corresponding tradeoff. We focus on work extraction protocols and show our results in two different frameworks: A collision model and a model where the Hamiltonian of the working system is controlled over time and the system can be brought into contact with a heat bath. Our results show that optimal processes are robust to noise and imperfections in small quantum systems, and can be achieved by a large set of interactions between system and bath.Comment: 12 pages + appendix; extended results; accepted in Quantu