Entanglement and symmetries in many-body quantum systems

As Schroedinger already recognised one century ago, entanglement is at the core of quantum mechanics. Nowadays it turns out to be the fundamental notion behind many quantum phenomena, from quantum algorithms to gravity, passing by critical phenomena and topological phases of matter, triggering unexpected connections between apparently far branches of physics. At the center of all these ideas, we find the (Rényi) entanglement entropies, which are powerful entanglement measures that provide fundamental insights into the investigated system or theory. This motivates the development of techniques for determining it, such as the replica trick: its implementation via the path-integral finds a wide place in this thesis. For example, we develop an efficient strategy to compute a generalised version of the Rényi entropies for all the eigenstates of a (1+1)-dimensional conformal field theory. This represents the starting point for a simulation scheme ideal to compute the entanglement in more generic (1+1)-dimensional quantum field theories (QFTs), e.g. after a quench in the sine-Gordon field theory. The study of entanglement also intertwines with another pillar of modern physics, i.e. symmetries and how their presence influences the properties of a system. Given the interest in this connection, this thesis addresses the question of how the entanglement splits into the different sectors of an internal symmetry. We approach the problem first in the QFT context, both for the free Dirac and complex scalar fields in two-dimensional spacetime, which have an abelian conserved charge, and systems having an internal Lie group symmetry to tackle the non-abelian case. Another typical framework in which we study the symmetry resolution of entanglement is lattice models, where different techniques can be exploited in order to derive exact results, ranging from the corner transfer matrix for gapped integrable systems to the connection between quadratic lattice Hamiltonians and their two-point correlation functions. The symmetry resolution also concerns other entanglement measures, namely, we analyse the behaviour of the operator entanglement, i.e. a key quantifier of the complexity of an operator, the symmetry-resolved mutual information, the effect of symmetries on entanglement negativity. The latter quantity is a genuine measure of quantum correlations in mixed states and a consistent part of the thesis is about this subject. For example, we study its time evolution after a quench and we provide an operatorial characterisation for entanglement in mixed states, which we dub negativity Hamiltonian

The Statistical Foundations of Entropy

In the last two decades, the understanding of complex dynamical systems underwent important conceptual shifts. The catalyst was the infusion of new ideas from the theory of critical phenomena (scaling laws, renormalization group, etc.), (multi)fractals and trees, random matrix theory, network theory, and non-Shannonian information theory. The usual Boltzmann–Gibbs statistics were proven to be grossly inadequate in this context. While successful in describing stationary systems characterized by ergodicity or metric transitivity, Boltzmann–Gibbs statistics fail to reproduce the complex statistical behavior of many real-world systems in biology, astrophysics, geology, and the economic and social sciences.The aim of this Special Issue was to extend the state of the art by original contributions that could contribute to an ongoing discussion on the statistical foundations of entropy, with a particular emphasis on non-conventional entropies that go significantly beyond Boltzmann, Gibbs, and Shannon paradigms. The accepted contributions addressed various aspects including information theoretic, thermodynamic and quantum aspects of complex systems and found several important applications of generalized entropies in various systems

Metrics and geodesics on fuzzy spaces

We study the fuzzy spaces (as special examples of noncommutative manifolds) with their quasicoherent states in order to find their pertinent metrics. We show that they are naturally endowed with two natural "quantum metrics" which are associated with quantum fluctuations of "paths". The first one provides the length the mean path whereas the second one provides the average length of the fluctuated paths. Onto the classical manifold associated with the quasicoherent state (manifold of the mean values of the coordinate observables in the state minimising their quantum uncertainties) these two metrics provides two minimising geodesic equations. Moreover, fuzzy spaces being not torsion free, we have also two different autoparallel geodesic equations associated with two different adiabatic regimes in the move of a probe onto the fuzzy space. We apply these mathematical results to quantum gravity in BFSS matrix models, and to the quantum information theory of a controlled qubit submitted to noises of a large quantum environment

Large consequences of quantum coherence in small systems

This thesis is concerned with the theoretical behaviour and interactions of quantum systems. It is composed of three main parts. We begin by investigating the correlations attainable when a bipartite quantum system undergoes unitary dynamics. The correlations are quantified by the quantum mutual information. We fully solve the problem for the smallest system of two qubits and present work towards solving the general case. The optimisation can be applied to thermodynamic scenarios, for example, heat exchange between two quantum systems. More specifically, we find bounds any negative heat flow from cold to hot, which can occur if the systems are initially correlated. We also present related applications such as a generalized collision model approach to thermal equilibrium, and a situation where a global Maxwell demon can play tricks on a local observer by reversing their local arrow of time. Experimental evidence suggests that biology may harness quantum effects to improve the efficiency of some of its processes. One such process is hydrogen transfer, catalysed by an enzyme called soybean lipoxygenase. The observed rates for this reaction strongly indicate that the hydrogen could be tunnelling through the energy barrier. We study this reaction by designing a qualitative model and find that our rates exhibit similar trends to those seen in experiments. The final part of the thesis is concerned with the quantum steering ellipsoid: a faithful, three-dimensional (3D) representation for the state of a two-qubit system. The steering ellipsoid is the set of states that Bob can collapse Alice's qubit to when he performs all possible measurements on his qubit. This formalism leads to numerous new features. We uncover a notion of incomplete steering of a separable state; geometric necessary and sufficient conditions for entanglement and discord; and a volume formula for the ellipsoid that identifies when steering is 3D, giving rise to a new type of correlation called "obesity".Open Acces

Few-Fermion Systems under a Matterwave Microscope

This thesis presents correlation measurements in two different few-fermion systems of ultracold 6Li atoms. The measurements have been performed with a new spatially and spin-resolved imaging method with single-atom sensitivity, with which we can probe coherences of the initial system as correlations in the momenta. First, we study attractively interacting atoms in a single microtrap, which serves as a basis for understanding the expansion dynamics of strongly interacting Fermi gases. We observe correlation features in the relative coordinate for different interaction strengths. We explain several of these features theoretically by calculating the initial interacting state in the microtrap and projecting it on a molecular bound state and scattering waves. Next, we study a small number of repulsively interacting particles in the ground state of a double-well potential. This system constitutes the fundamental building block of the Hubbard model. We observe interference patterns in the coordinates of the individual particles and in their relative coordinates. From the amplitude and phase of these patterns, we extract off-diagonal density matrix elements of the state, which we use to directly show coherence and entanglement in our system

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions