This topical review article gives an overview of the interplay between quantum information theory and thermodynamics of quantum systems. We focus on several trending topics including the foundations of statistical mechanics, resource theories, entanglement in thermodynamic settings, fluctuation theorems and thermal machines. This is not a comprehensive review of the diverse field of quantum thermodynamics; rather, it is a convenient entry point for the thermo-curious information theorist. Furthermore this review should facilitate the unification and understanding of different interdisciplinary approaches emerging in research groups around the world. CONTENTS

This work is concerned with thermal quantum states of Hamiltonians on spin and fermionic lattice systems with short range interactions. We provide results leading to a local definition of temperature, thereby extending the notion of "intensivity of temperature" to interacting quantum models. More precisely, we derive a perturbation formula for thermal states. The influence of the perturbation is exactly given in terms of a generalized covariance. For this covariance, we prove exponential clustering of correlations above a universal critical temperature that upper bounds physical critical temperatures such as the Curie temperature. As a corollary, we obtain that above the critical temperature, thermal states are stable against distant Hamiltonian perturbations. Moreover, our results imply that above the critical temperature, local expectation values can be approximated efficiently in the error and the system size.

In this work, we show how Gibbs or thermal states appear dynamically in closed quantum many-body systems, building on the program of dynamical typicality. We introduce a novel perturbation theorem for physically relevant weak system-bath couplings that is applicable even in the thermodynamic limit. We identify conditions under which thermalization happens and discuss the underlying physics. Based on these results, we also present a fully general quantum algorithm for preparing Gibbs states on a quantum computer with a certified runtime and error bound. This complements quantum Metropolis algorithms, which are expected to be efficient but have no known runtime estimates and only work for local Hamiltonians.

We review some of the recent progress on the study of entropy of entanglement in many-body quantum systems. Emphasis is placed on the scaling properties of entropy for one-dimensional multi-partite models at quantum phase transitions and, more generally, on the concept of area law. We also briefly describe the relation between entanglement and the presence of impurities, the idea of particle entanglement, the evolution of entanglement along renormalization group trajectories, the dynamical evolution of entanglement and the fate of entanglement along a quantum computation.

We study the existence of absolutely maximally entangled (AME) states in quantum mechanics and its applications to quantum information. AME states are characterized by being maximally entangled for all bipartitions of the system and exhibit genuine multipartite entanglement. With such states, we present a novel parallel teleportation protocol which teleports multiple quantum states between groups of senders and receivers. The notable features of this protocol are that (i) the partition into senders and receivers can be chosen after the state has been distributed, and (ii) one group has to perform joint quantum operations while the parties of the other group only have to act locally on their system. We also prove the equivalence between pure state quantum secret sharing schemes and AME states with an even number of parties. This equivalence implies the existence of AME states for an arbitrary number of parties based on known results about the existence of quantum secret sharing schemes. PACS numbers:Introduction. Entanglement is at the core of the power of quantum information processing and has been extensively studied for few qubits. The classification of entanglement classes for three and four qubits is well understood [1][2][3][4][5][6][7] and canonical forms of pure states under local unitary transformations of each local Hilbert space have also been analyzed [6,8,9]. As the number of local quantum degrees of freedom increases, our understanding of entanglement gets poorer. The number of independent invariants that classify entanglement grows exponentially and it is unclear which purpose each category of entanglement serves [10,11]. In recent years, there has been an important progress in the classification of the maximally multipartite entangled states composed of qubits [7,[12][13][14][15]. Nevertheless, a complete understanding of the structure, classification and usefulness of quantum states with the largest possible entanglement for arbitrary dimension is still missing. Another motivation for studying multipartite entanglement is its connection to other apparently unrelated areas of physics, like string theory and black-holes [16,17].Quantum teleportation is one of the most intriguing utilizations of entanglement. It allows distant parties, who share a resource of entanglement, to transport a quantum state from one party to the other by only exchanging classical information and using up said entanglement. The first proposal of such a protocol used the resource of bipartite entanglement between two parties [18]. Later teleportation protocols using genuine multipartite entanglement between more than two parties were proposed theoretically for four qubit entanglement [19], and experimentally in the form of open-destination teleportation for five qubits [20].This manuscript is devoted to initiate the study of a class of states with genuine multipartite entanglement. These states, which we call absolutely maximally entangled (AME) states, are defined as having the strict maximal entanglement in all

The dynamics of quantum phase transitions pose one of the most challenging problems in modern many-body physics. Here, we study a prototypical example in a clean and well-controlled ultracold atom setup by observing the emergence of coherence when crossing the Mott insulator to superfluid quantum phase transition. In the 1D Bose-Hubbard model, we find perfect agreement between experimental observations and numerical simulations for the resulting coherence length. We, thereby, perform a largely certified analog quantum simulation of this strongly correlated system reaching beyond the regime of free quasiparticles. Experimentally, we additionally explore the emergence of coherence in higher dimensions, where no classical simulations are available, as well as for negative temperatures. For intermediate quench velocities, we observe a power-law behavior of the coherence length, reminiscent of the Kibble-Zurek mechanism. However, we find nonuniversal exponents that cannot be captured by this mechanism or any other known model. ultracold atoms | optical lattice | Mott insulator | nonequilibrium dynamics | quantum simulation P hase transitions are ubiquitous but rather intricate phenomena, and it took until the late 19th century until a first theory of classical phase transitions was established. Quantum phase transitions (QPTs) are marked by sudden drastic changes in the nature of the ground state on varying a parameter of the Hamiltonian. They constitute one of the most intriguing frontiers of modern quantum many-body and condensed matter physics (1-4). Although it is typically possible to adiabatically follow the slowly changing ground state in a gapped phase, where the lowest excitation is separated from the ground state by a finite energy, these spectral gaps usually close at a QPT. Because the correlation length simultaneously diverges, adiabaticity is bound to break down, and several important questions emerge. How does a state dynamically evolve across the QPT (i.e., how does the transition literally happen?)? To what extent can the static ground state of a gapless phase be prepared in a realistic finitetime experiment? When entering a critical phase associated with an infinite correlation length, such as superfluid or ferromagnetic order, at what rate and by what mechanism will these correlations build up? Despite the fundamental importance of these questions, satisfactory answers have not been identified so far. Although the intrinsic complexity of the underlying nonintegrable models hinders numerical studies in most cases, the progress in the field of ultracold atoms now enables quantitative experiments in clean, well-isolated, and highly controllable systems.Here, we study the quantitative dynamics of a transition into a gapless, superfluid phase in the regime of short and intermediate quench times, finding complex behavior outside the scope of any current theoretical model. As a prototypical many-body system with a QPT, we use the transition from a Mott insulator to a superfluid in the Bose-Hubbard model (...

Absolutely maximally entangled (AME) states are those multipartite quantum states that carry absolute maximum entanglement in all possible bipartitions. AME states are known to play a relevant role in multipartite teleportation, in quantum secret sharing, and they provide the basis novel tensor networks related to holography. We present alternative constructions of AME states and show their link with combinatorial designs. We also analyze a key property of AME states, namely, their relation to tensors, which can be understood as unitary transformations in all of their bipartitions. We call this property multiunitarity.

Entanglement entropy obeys area law scaling for typical physical quantum systems. This may naively be argued to follow from locality of interactions. We show that this is not the case by constructing an explicit simple spin chain Hamiltonian with nearest neighbor interactions that presents an entanglement volume scaling law. This non-translational model is contrived to have couplings that force the accumulation of singlet bonds across the half chain. Our result is complementary to the known relation between non-translational invariant, nearest neighbor interacting Hamiltonians and QMA complete problems.