Fabio Anza

The unitary dynamics of isolated quantum systems does not allow a pure state to thermalize. Because of that, if an isolated quantum system equilibrates, it will do so to the predictions of the so-called "diagonal ensemble" r DE. Building on the intuition provided by Jaynes' maximum entropy principle, in this paper we present a novel technique to generate progressively better approximations to r DE. As an example, we write down a hierarchical set of ensembles which can be used to describe the equilibrium physics of small isolated quantum systems, going beyond the "thermal ansatz" of Gibbs ensembles.

We consider a closed region R of 3d quantum space modeled by SU (2) spin-networks. Using the concentration of measure phenomenon we prove that, whenever the ratio between the boundary ∂R and the bulk edges of the graph overcomes a finite threshold, the state of the boundary is always thermal, with an entropy proportional to its area. The emergence of a thermal state of the boundary can be traced back to a large amount of entanglement between boundary and bulk degrees of freedom. Using the dual geometric interpretation provided by loop quantum gravity, we interprete such phenomenon as a pre-geometric analogue of Thorne's "Hoop conjecture", at the core of the formation of a horizon in General Relativity.

Under unitary time evolution, expectation values of physically reasonable observables often evolve towards the predictions of equilibrium statistical mechanics. The eigenstate thermalization hypothesis (ETH) states that this is also true already for individual energy eigenstates. Here we aim at elucidating the emergence of ETH for observables that can realistically be measured due to their high degeneracy, such as local, extensive or macroscopic observables. We bisect this problem into two parts, a condition on the relative overlaps and one on the relative phases between the eigenbases of the observable and Hamiltonian. We show that the relative overlaps are completely unbiased for highly degenerate observables and demonstrate that unless relative phases conspire to cumulative effects this makes such observables verify ETH. Through connecting the degeneracy of observables and entanglement of the energy eigenstates this result elucidates potential pathways to equilibration in a fully general way. "Pure state quantum statistical mechanics"[1-5] aims at understanding under which conditions the use of tools from statistical mechanics can be justified based on the first principles of standard quantum mechanics with as few extra assumptions as possible. To explain the emergence of thermalization it combines three approaches: Typicality arguments [6-13], the dynamical equilibration approach [14-21] and the Eigenstate Thermalization Hypothesis (ETH) [22-34]. According to the first one, systems appear to be in equilibrium because, in a precise sense, most states are in equilibrium. Alternatively, according to the second approach apparent equilibration of observables and whole subsystems emerges because initial states of large many-body systems overlap with many energy eigenstates and therefore explore a large part of Hilbert space during their evolution, almost all the while being almost indistinguishable from a static equilibrium state. ETH, on the other hand, is a hypothesis about properties of individual eigenstates of sufficiently complicated quantum many-body systems which was suggested by various results in quantum chaos theory and it ad-duces the appearance of thermalization during such equi-libration to an underlying chaotic behavior. The basic idea is that, for large system sizes and in sufficiently complicated quantum many-body systems, the energy eigen-states can be so entangled that when we look at their overlaps with the basis of a physical observable they can be effectively described by random variables. If the ETH is fulfilled, it guarantees thermalization whenever equi-libration happens because of the mechanisms described above. Depending on how broad one wants the class of initial states that thermalize to be, the fulfillment of the ETH is also a necessary criterion for thermalization [5, 35]. The ETH is sometimes criticized for its lack of pre-dictive power, as it leaves open at least three important questions: what precisely are "physical observables"; what makes a system "sufficiently complicated" to expect that ETH applies; how long will it take for such observ-ables to reach thermal expectation values [18, 21]. For this reason, a lot of effort has been focused on numerical investigations that validate the ETH in specific Hamilto-nian models and for various observables, often including local ones. The ETH is generally found to hold in non-integrable systems that are not many-body localized and equilibration towards thermal expectation values usually happens on reasonable times scales [18, 20, 21, 34]. Recently [36] it has been shown that for any Hamilto-nian there is always a large number of observables which satisfy ETH. They have been dubbed "Hamiltonian Unbi-ased Observables" (HUO) and admit an algorithmic construction. Unfortunately this still leaves open when concrete physically relevant observables satisfy the ETH. In this letter we make progress in this direction. Building on the connection between HUOs and ETH, we present a theorem which can be used as a tool to investigate the emergence of ETH. In order to show how it can be used, we present three applications: local observables, extensive observables, and macro-observables. We will give precise definitions for each of them later. The paper is organized as follows. First we setup the notation, recall different formulations of the ETH and clarify which one we will be using throughout the paper. We continue with a brief digression on physical observables and degeneracies and recall the concepts of Hamiltonian unbiased basis and observables. We then present our main result, which elucidates the question under which conditions highly degenerate observables are HUO and discuss consequences of it for local observ-ables, extensive observables, and a certain type of macro

A crucial point in statistical mechanics is the definition of the notion of thermal equilibrium, which can be given as the state that maximises the von Neumann entropy, under the validity of some constraints. Arguing that such a notion can never be experimentally probed, in this paper we propose a new notion of thermal equilibrium, focused on observables rather than on the full state of the quantum system. We characterise such notion of thermal equilibrium for an arbitrary observable via the maximisation of its Shannon entropy and we bring to light the thermal properties that it heralds. The relation with Gibbs ensembles is studied and understood. We apply such a notion of equilibrium to a closed quantum system and show that there is always a class of observables which exhibits thermal equilibrium properties and we give a recipe to explicitly construct them. Eventually, an intimate connection with the Eigenstate Thermalisation Hypothesis is brought to light. To understand under which conditions thermodynamics emerges from the microscopic dynamics is the ultimate goal of statistical mechanics. However, despite the fact that the theory is more than 100 years old, we are still discussing its foundations and its regime of applicability. The ordinary way in which thermal equilibrium properties are obtained, in statistical mechanics, is through a complete characterisation of the thermal form of the state of the system. One way of deriving such form is by using Jaynes principle 1-4 , which is the constrained maximisation of von Neumann entropy S vN = − Trρ logρ. Jaynes showed that the unique state that maximises S vN (compatibly with the prior information that we have on the system) is our best guess about the state of the system at the equilibrium. The outcomes of such procedure are the so-called Gibbs ensembles. In the following we argue that such a notion of thermal equilibrium, de facto is not experimentally testable because it gives predictions about all possible observables of the system, even the ones which we are not able to measure. To overcome this issue, we propose a weaker notion of thermal equilibrium, specific for a given observable. The issue is particularly relevant for the so-called "Pure states statistical mechanics" 5-19 , which aims to understand how and in which sense thermal equilibrium properties emerge in a closed quantum system, under the assumption that the dynamic is unitary. In the last fifteen years we witnessed a revival of interest in these questions , mainly due to remarkable progresses in the experimental investigation of isolated quantum systems 20-25. The high degree of manipulability and isolation from the environment that we are able to reach nowadays makes possible to experimentally investigate such questions and to probe the theoretical predictions. The starting point of Jaynes' derivation of statistical mechanics is that S vN is a way of estimating the uncertainty that we have about which pure state the system inhabits. Unfortunately we know from quantum information theory that it does not address all kind of ignorance we have about the system. Indeed, it is not the entropy of an observable (though the state is observable); its conceptual meaning is not tied to something that we can measure. This issue is intimately related with the way we acquire information about a system, i.e. via measurements. The process of measuring an observable on a quantum system allows to probe only the diagonal part of the density matrix λ ρ λ i i , when this is written in the observable eigenbasis λ { } i. For such a reason, from the experimental point of view, it is not possible to assess whether a many-body quantum system is at thermal equilibrium (e.g. Gibbs state ρ G): the number of observables needed to probe all the density matrix elements is too big. In any experimentally reasonable situation we have access only to a few (sometimes just one or two) observables. It is 1 tomic ann aser sicss arennon aaoratorr niiersitt of OOforr arrs oaa OOforr O1 333. Centre for uantum eccnooooiess ationaa niiersitt of innaporee 1177433 innapore. 3 Department of Physics, National niiersitt of innapore cience Driie 3 1171 innapore. 4 Center for Quantum Information, Institute for Interiscippinar Information ciences sinua niiersit 1000844 eiiin ina. * These authors contributed eeua to tis wor. orresponence an reeuests for materias souu e aressee to .. emai: faio.ana pppsics.oo.ac.uuu receiiee: 13 Octooer 016 acceptee: 31 anuarr 017 Puuuissee: 07 arcc 017 OPEN

In this work we extend the so-called typicality approach, originally formulated in statistical mechanics contexts, to SU (2) invariant spin network states. Our results do not depend on the physical interpretation of the spin-network, however they are mainly motivated by the fact that spin-network states can describe states of quantum geometry, providing a gauge-invariant basis for the kinematical Hilbert space of several background independent approaches to quantum gravity. The first result is, by itself, the existence of a regime in which we show the emergence of a typical state. We interpret this as the prove that, in that regime there are certain (local) properties of quantum geometry which are "universal". Such set of properties is heralded by the typical state, of which we give the explicit form. This is our second result. In the end, we study some interesting properties of the typical state, proving that the area-law for the entropy of a surface must be satisfied at the local level, up to logarithmic corrections which we are able to bound.

We exploit the tripartite negativity to study the thermal correlations in a tripartite system, that is, the three outer spins interacting with the central one in a spin–star system. We analyse the dependence of such correlations on the homogeneity of interactions, starting from the case where central–outer spin interactions are identical and then focusing on the case where the three coupling constants are different. We single out some important differences between the negativity and the concurrence.