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We develop the theory of local operations and classical communication (LOCC) for bipartite quantum systems represented by commuting von Neumann algebras. Our central result is the extension of Nielsen's Theorem, stating that the LOCC ordering of bipartite pure states is equivalent to the majorization of their restrictions, to arbitrary factors. As a consequence, we find that in bipartite system modeled by commuting factors in Haag duality, a) all states have infinite single-shot entanglement if and only if the local factors are not of type I, b) type III factors are characterized by LOCC transitions of arbitrary precision between any two pure states, and c) the latter holds even without classical communication for type III$_{1}$ factors. In the case of semifinite factors, the usual construction of pure state entanglement monotones carries over. Together with recent work on embezzlement of entanglement, this gives a one-to-one correspondence between the classification of factors into types and subtypes and operational entanglement properties. In the appendix, we provide a self-contained treatment of majorization on semifinite von Neumann algebras and $\sigma$-finite measure spaces., Comment: 35+13+5 pages, 1 figure; v2: improved presentation, added references

Embezzlement of entanglement refers to the task of extracting entanglement from an entanglement resource via local operations and without communication while perturbing the resource arbitrarily little. Recently, the existence of embezzling states of bipartite systems of type III von Neumann algebras was shown. However, both the multipartite case and the precise relation between embezzling states and the notion of embezzling families, as originally defined by van Dam and Hayden, was left open. Here, we show that finite-dimensional approximations of multipartite embezzling states form multipartite embezzling families. In contrast, not every embezzling family converges to an embezzling state. We identify an additional consistency condition that ensures that an embezzling family converges to an embezzling state. This criterion distinguishes the embezzling family of van Dam and Hayden from the one by Leung, Toner, and Watrous. The latter generalizes to the multipartite setting. By taking a limit, we obtain a multipartite system of commuting type III$_1$ factors on which every state is an embezzling state. We discuss our results in the context of quantum field theory and quantum many-body physics. As open problems, we ask whether vacua of relativistic quantum fields in more than two spacetime dimensions are multipartite embezzling states and whether multipartite embezzlement allows for an operator-algebraic characterization., Comment: 34 pages, 2 figures, comments welcome

Universal embezzlers are bipartite quantum systems from which any entangled state may be extracted to arbitrary precision using local operations while perturbing the state of the system arbitrarily little. Here, we show that universal embezzlers are ubiquitous in many-body physics: The ground state sector of every local, translation-invariant, and critical free-fermionic many-body system on a one-dimensional lattice is a universal embezzler if bi-partitioned into two half-chains. The same property holds in locally-interacting, dual spin chains via the Jordan-Wigner transformation. Universal embezzlement manifests already for finite system sizes, not only in the thermodynamic limit: For any finite error and any targeted entangled state, a finite length of the chain is sufficient to embezzle said state within the given error. On a technical level, our main result establishes that the half-chain observable algebras associated with ground state sectors of the given models are type III$_1$ factors., Comment: 27 pages, 1 figure, comments welcome; minor corrections and improvements, results unaltered (v2)

We study the quantum information theoretic task of embezzlement of entanglement in the setting of von Neumann algebras. Given a shared entangled resource state, this task asks to produce arbitrary entangled states using local operations without communication while perturbing the resource arbitrarily little. We quantify the performance of a given resource state by the worst-case error. States for which the latter vanishes are 'embezzling states' as they allow to embezzle arbitrary entangled states with arbitrarily small error. The best and worst performance among all states defines two algebraic invariants for von Neumann algebras. The first invariant takes only two values. Either it vanishes and embezzling states exist, which can only happen in type III, or no state allows for nontrivial embezzlement. In the case of factors not of finite type I, the second invariant equals the diameter of the state space. This provides a quantitative operational interpretation of Connes' classification of type III factors within quantum information theory. Type III$_1$ factors are 'universal embezzlers' where every state is embezzling. Our findings have implications for relativistic quantum field theory, where type III algebras naturally appear. For instance, they explain the maximal violation of Bell inequalities in the vacuum. Our results follow from a one-to-one correspondence between embezzling states and invariant probability measures on the flow of weights. We also establish that universally embezzling ITPFI factors are of type III$_1$ by elementary arguments., Comment: See arXiv:2401.07292 for an overview article; 68 pages + 1 table + 1 figure; comments welcome; v3: resolved open problems

Embezzlement of entanglement refers to the counterintuitive possibility of extracting entangled quantum states from a reference state of an auxiliary system (the "embezzler") via local quantum operations while hardly perturbing the latter. We uncover a deep connection between the operational task of embezzling entanglement and the mathematical classification of von Neumann algebras. Our result implies that relativistic quantum fields are universal embezzlers: Any entangled state of any dimension can be embezzled from them with arbitrary precision. This provides an operational characterization of the infinite amount of entanglement present in the vacuum state of relativistic quantum field theories., Comment: Overview paper accompanying the technical manuscript arXiv:2401.07299; v2: improved presentation and added an End Matter including a short introduction to von Neumann algebraic Quantum Information 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 figures

Catalysts open up new reaction pathways that 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 returns to its initial state at the end of the process. In this review, a comprehensive overview of the concept of catalysis in quantum information science is presented and its applications in various physical contexts are discussed., Comment: Review paper; Close to published version

A set of classical or quantum states is equivalent to another one if there exists a pair of classical or quantum channels mapping either set to the other one. For dichotomies (pairs of states), this is closely connected to (classical or quantum) R\'enyi divergences (RD) and the data-processing inequality: If a RD remains unchanged when a channel is applied to the dichotomy, then there is a recovery channel mapping the image back to the initial dichotomy. Here, we prove for classical dichotomies that equality of the RDs alone is already sufficient for the existence of a channel in any of the two directions and discuss some applications. In the quantum case, all families of quantum RDs are seen to be insufficient because they cannot detect anti-unitary transformations. Thus, including anti-unitaries, we pose the problem of finding a sufficient family. It is shown that the Petz and maximal quantum RD are still insufficient in this more general sense and we provide evidence for sufficiency of the minimal quantum RD. As a side result of our techniques, we obtain an infinite list of inequalities fulfilled by the classical, the Petz quantum, and the maximal quantum RDs. These inequalities are not true for the minimal quantum RDs. Our results further imply that any sufficient set of conditions for state transitions in the resource theory of athermality must be able to detect time-reversal., Comment: Comments welcome, 40 pages, major changes, modified conjecture

Catalysts are quantum systems that open up dynamical pathways between quantum states which are otherwise inaccessible under a given set of operational restrictions while, at the same time, they do not change their quantum state. We here consider the restrictions imposed by symmetries and conservation laws, where any quantum channel has to be covariant with respect to the unitary representation of a symmetry group, and present two results. First, for an exact catalyst to be useful, it has to build up correlations to either the system of interest or the degrees of freedom dilating the given process to covariant unitary dynamics. This explains why catalysts in pure states are useless. Second, if a quantum system ("reference frame") is used to simulate to high precision unitary dynamics (which possibly violates the conservation law) on another system via a global, covariant quantum channel, then this channel can be chosen so that the reference frame is approximately catalytic. In other words, a reference frame that simulates unitary dynamics to high precision degrades only very little., Comment: 12+1 pages. Accepted for Publication at Quantum - the open journal for quantum science

When a generic quantum system is prepared in a simple initial condition, it typically equilibrates toward a state that can be described by a thermal ensemble. A known exception are localized systems which are non-ergodic and do not thermalize, however local observables are still believed to become stationary. Here we demonstrate that this general picture is incomplete by constructing product states which feature periodic high-fidelity revivals of the full wavefunction and local observables that oscillate indefinitely. The system neither equilibrates nor thermalizes. This is analogous to the phenomenon of weak ergodicity breaking due to many-body scars and challenges aspects of the current MBL phenomenology, such as the logarithmic growth of the entanglement entropy. To support our claim, we combine analytic arguments with large-scale tensor network numerics for the disordered Heisenberg chain. Our results hold for arbitrarily long times in chains of 160 sites up to machine precision., Comment: 11+3 pages, 6+3 figures; Close to published version

I show that if a finite-dimensional density matrix has strictly smaller von Neumann entropy than a second one of the same dimension (and the rank is not bigger), then sufficiently (but finitely) many tensor-copies of the first density matrix majorize a density matrix whose single-body marginals are all exactly equal to the second density matrix. This implies an affirmative solution of the exact catalytic entropy conjecture (CEC) introduced by Boes et al. [PRL 122, 210402 (2019)]. Both the Lemma and the solution to the CEC transfer to the classical setting of finite-dimensional probability vectors (with permutations of entries instead of unitary transformations for the CEC)., Comment: v2: ~3.5 + 1 pages; Improved presentation, additional appendix; final version for publication in Quantum

It was recently observed that in certain thermalizing many-body systems, measuring the complement of a subsystem that thermalized to infinite temperature in a suitable orthonormal basis gives rise to approximate quantum state k-designs as post-measurement states on the subsystem. We prove that this emergence of approximate k-designs holds true in every large system where some small subsystem is close to being maximally mixed. On a technical level we show that any high-dimensional purification of an approximately maximally mixed state induces an approximate quantum state k-design when measured in a suitable orthonormal basis. Moreover, we show that this is true with overwhelming probability for measurement bases chosen uniformly at random from the Haar measure., Comment: 4+6 pages

We observe that the many-body eigenstates of any quadratic, fermionic Hamiltonian with sublattice symmetry have quantized entanglement entropies between the sublattices: the entanglement comes in multiple singlets. Moreover, such systems always have a ground state that is maximally entangled between the two sublattices. In fact we also show that under the same assumptions there always exists a (potentially distinct) basis of energy eigenstates that do not conserve the particle number in which each energy eigenstate is maximally entangled between the sublattices. No additional properties, such as translation invariance, are required. We also show that the quantization of ground state entanglement may persist when interactions are introduced., Comment: 5+4 pages, 4 figures; v2: added results for interacting model, improved presentation, close to published version

I show that non-decreasing entropy provides a necessary and sufficient condition to convert the state of a physical system into a different state by a reversible transformation that acts on the system of interest and a further "catalyst" whose state has to remain invariant exactly in the transition. This statement is proven both in the case of finite-dimensional quantum mechanics, where von~Neumann entropy is the relevant entropy, and in the case of systems whose states are described by probability distributions on finite sample spaces, where Shannon entropy is the relevant entropy. The results give an affirmative resolution to the (approximate) "catalytic entropy conjecture" introduced by Boes et al. [PRL 122, 210402 (2019)]. They provide a complete single-shot characterization without external randomness of von Neumann entropy and Shannon entropy. I also compare the results to the setting of phenomenological thermodynamics and show how they can be used to obtain a quantitative single-shot characterization of Gibbs states in quantum statistical mechanics., Comment: 5+5 pages; 3+1 figures. Comments welcome!; v2: added comparison to thermodynamics and application to quantum statistical mechanics, expanded appendix, minor corrections, close to published version

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 updated

The variance of (relative) surprisal, also known as varentropy, so far mostly plays a role in information theory as quantifying the leading order corrections to asymptotic i.i.d.~limits. Here, we comprehensively study the use of it to derive single-shot results in (quantum) information theory. We show that it gives genuine sufficient and necessary conditions for approximate state-transitions between pairs of quantum states in the single-shot setting, without the need for further optimization. We also clarify its relation to smoothed min- and max-entropies, and construct a monotone for resource theories using only the standard (relative) entropy and variance of (relative) surprisal. This immediately gives rise to enhanced lower bounds for entropy production in random processes. We establish certain properties of the variance of relative surprisal which will be useful for further investigations, such as uniform continuity and upper bounds on the violation of sub-additivity. Motivated by our results, we further derive a simple and physically appealing axiomatic single-shot characterization of (relative) entropy which we believe to be of independent interest. We illustrate our results with several applications, ranging from interconvertibility of ergodic states, over Landauer erasure to a bound on the necessary dimension of the catalyst for catalytic state transitions and Boltzmann's H-theorem., Comment: v3: published version

We prove that all R\'enyi entanglement entropies of spin-chains described by generic (gapped), translational invariant matrix product states (MPS) are extensive for disconnected sub-systems: All R\'enyi entanglement entropy densities of the sub-system consisting of every k-th spin are non-zero in the thermodynamic limit if and only if the state does not converge to a product state in the thermodynamic limit. Furthermore, we provide explicit lower bounds to the entanglement entropy in terms of the expansion coefficient of the transfer operator of the MPS and spectral properties of its fixed point in canonical form. As side-result we obtain a lower bound for the expansion coefficient and singular value distribution of a primitve quantum channel in terms of its Kraus-rank and entropic properties of its fixed-point. For unital quantum channels this yields a very simple lower bound on the distribution of singular values and the expansion coefficient in terms of the Kraus-rank. Physically, our results are motivated by questions about equilibration in many-body localized systems, which we review., Comment: 18+2 pages, 2 figures; v2: fixed problem with figures; Comments welcome!

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 presentation

To make progress in science, we often build abstract representations of physical systems that meaningfully encode information about the systems. The representations learnt by most current machine learning techniques reflect statistical structure present in the training data; however, these methods do not allow us to specify explicit and operationally meaningful requirements on the representation. Here, we present a neural network architecture based on the notion that agents dealing with different aspects of a physical system should be able to communicate relevant information as efficiently as possible to one another. This produces representations that separate different parameters which are useful for making statements about the physical system in different experimental settings. We present examples involving both classical and quantum physics. For instance, our architecture finds a compact representation of an arbitrary two-qubit system that separates local parameters from parameters describing quantum correlations. We further show that this method can be combined with reinforcement learning to enable representation learning within interactive scenarios where agents need to explore experimental settings to identify relevant variables., Comment: 24 pages, 13 figures

We derive general results relating revivals in the dynamics of quantum many-body systems to the entanglement properties of energy eigenstates. For a D-dimensional lattice system of N sites initialized in a low-entangled and short-range correlated state, our results show that a perfect revival of the state after a time at most poly(N) implies the existence of "quantum many-body scars", whose number grows at least as the square root of N up to poly-logarithmic factors. These are energy eigenstates with energies placed in an equally-spaced ladder and with R\'enyi entanglement entropy scaling as log(N) plus an area law term for any region of the lattice. This shows that quantum many-body scars are a necessary condition for revivals, independent of particularities of the Hamiltonian leading to them. We also present results for approximate revivals, for revivals of expectation values of observables and prove that the duration of revivals of states has to become vanishingly short with increasing system size., Comment: 7+8 pages, 3 figures, new author and new results on Renyi entropies in v2

Fluctuation theorems impose constraints on possible work extraction probabilities in thermodynamical processes. These constraints are stronger than the usual second law, which is concerned only with average values. Here, we show that such constraints, expressed in the form of the Jarzysnki equality, can be bypassed if one allows for the use of catalysts - additional degrees of freedom that may become correlated with the system from which work is extracted, but whose reduced state remains unchanged so that they can be re-used. This violation can be achieved both for small systems but also for macroscopic many-body systems and leads to positive work extraction per particle with finite probability from macroscopic states in equilibrium. In addition to studying such violations for a single system, we also discuss the scenario in which many parties use the same catalyst to induce local transitions. We show that there exist catalytic processes that lead to highly correlated work distributions, expected to have implications for stochastic and quantum thermodynamics., Comment: 16 pages (6+10), published version

The area law conjecture states that the entanglement entropy of a region of space in the ground state of a gapped, local Hamiltonian only grows like the surface area of the region. We show that, for any quantum state that fulfills an area law, the reduced quantum state of a region of space can be unitarily compressed into a thickened surface of the region. If the interior of the region is lost after this compression, the full quantum state can be recovered to high precision by a quantum channel only acting on the thickened surface. The thickness of the boundary scales inversely proportional to the error for arbitrary spin systems and logarithmically with the error for quasi-free bosonic systems. Our results can be interpreted as a single-shot operational interpretation of the area law. The result for spin systems follows from a simple inequality showing that probability distributions with low entropy can be approximated by distributions with small support, which we believe to be of independent interest. We also discuss an emergent approximate correspondence between bulk and boundary operators and the relation of our results to tensor network states., Comment: 4+5 pages, 1 figure; v2: improved presentation, removed erroneous lemma from appendix (results unchanged), close to published version

Despite the success of neural networks at solving concrete physics problems, their use as a general-purpose tool for scientific discovery is still in its infancy. Here, we approach this problem by modelling a neural network architecture after the human physical reasoning process, which has similarities to representation learning. This allows us to make progress towards the long-term goal of machine-assisted scientific discovery from experimental data without making prior assumptions about the system. We apply this method to toy examples and show that the network finds the physically relevant parameters, exploits conservation laws to make predictions, and can help to gain conceptual insights, e.g. Copernicus' conclusion that the solar system is heliocentric., Comment: 4 pages main text + 11 pages appendix, changes since v2: improved references and presentation

One of the outstanding problems in non-equilibrium physics is to precisely understand when and how physically relevant observables in many-body systems equilibrate under unitary time evolution. General equilibration results show that equilibration is generic provided that the initial state has overlap with sufficiently many energy levels. But results not referring to typicality which show that natural initial states actually fulfill this condition are lacking. In this work, we present stringent results for equilibration for systems in which Renyi entanglement entropies in energy eigenstates with finite energy density are extensive for at least some, not necessarily connected, sub-system. Our results reverse the logic of common arguments, in that we derive equilibration from a weak condition akin to the eigenstate thermalization hypothesis, which is usually attributed to thermalization in systems that are assumed to equilibrate in the first place. We put the findings into the context of studies of many-body localization and many-body scars., Comment: 6+8 pages; v3: improved presentation, close to published version

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 Quantum

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 figures

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., Comment: 7 pages

The third law of thermodynamics in the form of the unattainability principle states that exact ground-state cooling requires infinite resources. Here we investigate the amount of non-equilibrium resources needed for approximate cooling. We consider as resource any system out of equilibrium, allowing for resources beyond the i.i.d. assumption and including the input of work as a particular case. We establish in full generality a sufficient and a necessary condition for cooling and show that for a vast class of non-equilibrium resources these two conditions coincide, providing a single necessary and sufficient criterion. Such conditions are expressed in terms of a single function playing a similar role for the third law to the one of the free energy for the second law. From a technical point of view we provide new results about concavity/convexity of certain Renyi-divergences, which might be of independent interest., Comment: Extended discussion on approximated and exact catalysts and models for the source of work. 22 pages. 2 Figures

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 non-trivial 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., Comment: Substantial rewrite of the previous version: "quantum source-channel codes". 6+2 pages, 4 figures