This thesis presents insights obtained on three questions in condensed matter physics via techniques in quantum information. ? The first topic deals with signatures of equilibration in a closed quantum system. Using the Loschmidt echo as a representative observable, that is derived from fidelity?a popular quantity in quantum information, and by studying it's long time statistics the role of quantum criticality in the equilibration dynamics of a closed system is analysed. While off?critical systems under a quantum quench are shown to equilibrate well, critical systems are shown to do so poorly signified by relatively large fluctuations of the echo around it?s long time average. ? The second topic deals with typicality of entanglement in the physical Hilbert space. Here, a general framework for studying statistical moments of physically relevant quantities in ensembles of quantum states generated by Local Random Quantum Circuits (LRQC) is outlined. These ensembles are constructed by finite?length random quantum circuits acting on the (hyper)edges of an underlying (hyper)graph structure. The latter are designed to encode for the locality structure associated with finite?time quantum evolutions generated by physical i.e. local Hamiltonians. Physical properties of typical states in these ensembles, in particular purity as a proxy of quantum entanglement is studied. The problem is formulated in terms of matrix elements of superoperators which depend on the graph structure, choice of probability measure over the local unitaries and circuit length. We consider different families of LRQCs and study their typical entanglement properties for finite?time as well as their asymptotic behavior. In particular, for a model of LRQC that resembles closely the Trotter scheme of discretizing quantum evolutions with local Hamiltonians, we find that the area law holds in average and that the volume law is a typical property (that is, it holds in average and the fluctuations around the average are vanishing for the large system) of physical states. The area law arises when the evolution time is O(1) with respect to the size L of the system, while the volume law arises as typical when the evolution time scales like O(L). ? The final topic deals with the perturbative response of the set of Re?nyi entropies of a subsystem when the entire system is in a state displaying some quantum order. The characteristic behavior of the entropies is shown to be able to identify topologically non?trivial and trivial phases in the case of quantum double models. The implications of the response towards the possibility of simulating the adiabatic evolution within a phase using the protocol of Local Operations and Classical Communications are discussed.
This thesis presents insights obtained on three questions in condensed matter physics via techniques in quantum information. ? The first topic deals with signatures of equilibration in a closed quantum system. Using the Loschmidt echo as a representative observable, that is derived from fidelity?a popular quantity in quantum information, and by studying it's long time statistics the role of quantum criticality in the equilibration dynamics of a closed system is analysed. While off?critical systems under a quantum quench are shown to equilibrate well, critical systems are shown to do so poorly signified by relatively large fluctuations of the echo around it?s long time average. ? The second topic deals with typicality of entanglement in the physical Hilbert space. Here, a general framework for studying statistical moments of physically relevant quantities in ensembles of quantum states generated by Local Random Quantum Circuits (LRQC) is outlined. These ensembles are constructed by finite?length random quantum circuits acting on the (hyper)edges of an underlying (hyper)graph structure. The latter are designed to encode for the locality structure associated with finite?time quantum evolutions generated by physical i.e. local Hamiltonians. Physical properties of typical states in these ensembles, in particular purity as a proxy of quantum entanglement is studied. The problem is formulated in terms of matrix elements of superoperators which depend on the graph structure, choice of probability measure over the local unitaries and circuit length. We consider different families of LRQCs and study their typical entanglement properties for finite?time as well as their asymptotic behavior. In particular, for a model of LRQC that resembles closely the Trotter scheme of discretizing quantum evolutions with local Hamiltonians, we find that the area law holds in average and that the volume law is a typical property (that is, it holds in average and the fluctuations around the average are vanishing for the large system) of physical states. The area law arises when the evolution time is O(1) with respect to the size L of the system, while the volume law arises as typical when the evolution time scales like O(L). ? The final topic deals with the perturbative response of the set of Re?nyi entropies of a subsystem when the entire system is in a state displaying some quantum order. The characteristic behavior of the entropies is shown to be able to identify topologically non?trivial and trivial phases in the case of quantum double models. The implications of the response towards the possibility of simulating the adiabatic evolution within a phase using the protocol of Local Operations and Classical Communications are discussed.
In this dissertation, we address several problems in condensed matter physics, statistical mechanics, and quantum chaos from the perspective of the quantum information-theoretic notion of state distinguishability, as measured though evaluation of the appropriate fidelity measure. ❧ We find that the fidelity susceptibility, a measure for the response of the fidelity to infinitesimal changes of the Hamiltonian parameters, serves as a useful tool for studying the effects of disorder on quantum phase transitions. We study in detail the phase diagram of the well-known one-dimensional random quantum XY chain. Through an analysis of the fidelity susceptibility’s finite-size scaling and statistics for this model, we find that the fidelity approach reflects large variation between average and typical behavior, may detect the presence of Griffiths phases, and indicates the modifications to the phase diagram due to disorder. ❧ For this same model, though without disorder, we evaluate the dynamics following a sudden Hamiltonian quench using the Loschmidt echo, a form of the fidelity that compares the initial state with the time-evolved state. Through a central limit theorem-type argument we find that the long-time statistics of the Loschmidt echo takes one of two universal forms, even at finite temperature, depending on the system’s proximity to the quantum critical lines. ❧ Finally, for a canonical model exhibiting quantum chaos, the hydrogen atom in a uniform external magnetic field, we evaluate the operator fidelity susceptibility (OFS) between unitary operators generating the dynamics. We find that the part of the OFS which corresponds to the variation of the Hamiltonian eigenvectors serves to indicate the transition from the regular perturbative and n-mixing regimes to the quantum chaotic regime.
Recently, the possibility of investigating single molecule, or single spin observables, as well as the necessity of a better understanding of the mechanisms underlying quantum dynamics in order to obtain nanoscale devices and nanostructered materials suitable for quantum computing tasks, have revived the interest in foundational aspects of quantum statistical mechanics. This thesis aims to give a contribution to this field by re-considering the statistical characterization of a quantum system at the light of some paradigmatic changes in our understanding of quantum theory which have taken place in the last two decades. In particular the impressive development of quantum information theory has changed the perceptions of quantum entanglement: for a long time it has been considered a somewhat paradoxical property of the matter at the atomic scale, but now it is regarded as an essential and ubiquitous phenomenon whose consequences are affecting the very macroscopic world that we experience. Still the decoherence program has brought out the importance of considering a quantum system together with its environment in order to clarify some key aspects of quantum dynamics. Thus, we start from the idea that quantum correlations are ubiquitous and somewhat uncontrollable in systems with many degrees of freedom which are typically considered in statistical mechanics. As a consequence we assume the standpoint that quantum statistical mechanics has not to be based on the underlying idea of a collection of many, independent quantum systems but rather it has to emerge at the level of a global wavefunction (pure state) which describes the system as well as its environment as a whole. In order to investigate the consequences of these assumptions we study the equilibrium distribution of an isolated quantum system. This is defined, in analogy with the ergodic foundations of classical statistical mechanics, on the basis of the time evolution of the quantum state. Then, we study the emergence of thermodynamic properties in a quantum system by studying the probability distribution of some function of interests, as the entropy and the equilibrium state of a subsystem, on Ensembles of Pure States. Such a probability distribution is derived from the geometry of the Hilbert space, and the theoretical tools suitable for its characterization are developed. On the one hand we perform a numerical sampling of the ensemble distributions by employing Monte Carlo techniques, on the other hand simpler analytical approximation of the geometrical distributions are derived by means of a maximum entropy principle. Model systems composed of an ensemble of spins are chosen to illustrate the salient features which emerge from the developed theoretical framework: the main point is that the Ensemble Distributions of “thermodynamic observables” (entropy or equilibrium state of a subsystem) are sharply peaked around a typical value. From the analysis it emerges that each of the overwhelming majority of the wavefunctions which has appreciable weight in the considered ensemble, is characterized by the same value of the “macroscopic” functions. This is a striking evidence of the “typicality” of these properties. In the essence, our impossibility to know the state of the system in detail does not matter, just for the remarkable fact that almost all quantum states behave essentially in the same way. By virtue of this typicality the study of the behaviour of the typical values of the thermodynamic function become meaningful. Notably, under certain conditions, one recovers the results of standard statistical mechanics, that is, the equilibrium average of the state of a subsystem can be cast in the Boltzmann canonical form at the temperature given by the usual thermodynamical relation . In the second part of the thesis we consider the dynamical aspects of the equilibrium state of a subsystem interacting with its environment. The fluctuations around the equilibrium average critically depends on the entanglement between the system and the environment and on the form of the interaction Hamiltonian. The connection between the dynamics of the fluctuations of an observable at the equilibrium and the relaxation toward the equilibrium from a “non typical” initial value is also investigated with the aid of simple model systems. The study presented in this thesis was partly motivated by a critical analysis of the statistical methods available for the theoretical modelling of magnetic resonance experiments. One of these, the Stochastic Liouville Equation, has been employed in a work completed during the first year of my Ph.D. program in order to interpret some feature of a two dimensional electron spin resonance experiment, [Fresch B., Frezzato D., Moro G. J., Kothe G., Freed J. H.; J. Phys. Chem. B., 110, 24238, (2006)].
In this thesis we explore several areas which lie between the traditional condensed matterphysics and the relatively more modern quantum information processing paradigms. Wemove from a single-body problem which displays chaotic behaviour, to two many-body systemswhich show a quantum phase transition and sufficient entanglement to perform complexquantum information processing, respectively.Firstly, we consider a set of operations which can easily be performed on a single ion in anion trap quantum computer, and associate this with an effective Hamiltonian. We determinethe classical and quantum bifurcation structure, and consider the physical signatures ofa bifurcation which may be measured in an experiment. Secondly, we consider a set ofoperations which may easily be performed on many ions in an ion trap quantum computer.We determine that the evolution of the system may be described by an effective Hamiltonian,and determine the nature of an effective quantum phase transition which may be experiencedby the system. We determine to what extent entanglement is present in this system, andhow even relatively small finite systems may give us insights into quantum phase transitions.Finally, in the many-body setting, we consider the use of teleportation as a characteristicof entanglement, and propose a teleportation scheme based on Bell basis measurement. Wedetermine that non-local, topological order parameters can be used to find the entanglementcontent necessary to perform a teleportation, and that we can understand the resourcesrequired for teleportation in terms of a quantum information paradigm that transcendstraditional local order parameters.In all three systems, we consider models which have “tunable” local and non-local interactions.Since the “quintessential feature of quantum mechanics” is the superposition ofnon-local states, it is natural to refer, time and again, to entanglement to variously describethe qualitative phenomenon of a system’s non-locality, its phase transition or its ability toperform a quantum information processing task. In each case, using a quantitative measureof entanglement, we are able to present concrete evidence for the essential role which entanglementplays in determining the physics we may observe from these quantum phenomenon.
Quantum entanglement has become the key notion bridging originally distinct fields of research over the last decade, namely, quantum information and computation, condensed matter physics, and quantum gravity. Previous studies on quantum entanglement have largely focused on the entanglement entropy, which quantifies the amount of entanglement. However, a natural question arises: is there additional information of a quantum state that is not captured by the entanglement entropy alone? For ground states of gapped Hamiltonians, this question has been answered in the affirmative. In this dissertation, I extend this idea to study highly entangled states typically having volume law entropy, and demonstrate that there is indeed much richer information on the complexity of a quantum state beyond the entanglement entropy. \n \nIn the first part, I study the entanglement spectrum of highly entangled states corresponding to highly excited eigenstates of non-integrable Hamiltonians, time-evolved states after a quantum quench with Hamiltonians exhibiting different dynamical phases, and random unitary circuits consisting of random braids of non-Abelian anyons. I demonstrate that the entanglement spectrum is able to capture the degree of randomness of a quantum state, which we call the entanglement complexity. In the context of scrambling, this quantifies the degree of randomness produced by scrambling beyond entropic diagnostics. \n \nOur understanding of quantum entanglement in condensed matter systems and high energy physics have largely benefited from the field of quantum computation. In the second part of the dissertation, I present two examples of novel platforms for quantum computation using state-of-the-art experimental technologies. I demonstrate how one can use hybrid quantum-classical architecture to solve computational problems based on an optimal variational ansatz of the evolution protocol. I also present a hierarchical architecture of constructing logical Majorana zero modes which can be used for demonstrating non-Abelian braiding statistics experimentally.
In the first part of this dissertation, we study the dynamics of isolated and clean quantum systems out of equilibrium. We initially address the Kibble-Zurek (KZ) problem of determining the dynamical evolution of a system close to its critical point under slow changes of a control parameter. We formulate a scaling limit in which the nonequilibrium behavior is universal and discuss the universal content. We then report computations of some scaling functions in model Gaussian and large-N problems. Next, we apply KZ scaling to topologically ordered systems with no local order parameter. In the examples of the Ising gauge theory and the SU(2)k phases of the Levin-Wen models, we observe a slow, coarsening dynamics for the string-net that underlies the physics of the topological phase at late times for ramps across transitions that reduce topological order. We conclude by studying quenches in the quantum O(N) model in the infinite N limit in varying spatial dimensions. Despite the failure to equilibrate owing to an infinite number of emergent conservation laws, the qualitative features of late time states following quenches is predicted by the equilibrium phase diagram. In the second part of this dissertation, we explore the relationship between entanglement and topological order in fractional quantum Hall (FQH) phases. In 2008, Li and Haldane conjectured that the entanglement spectrum (ES), a presentation of the Schmidt values of a real space cut, reflects the energy spectrum of the FQH chiral edge. Specifically, both spectra should have the same quasi-degeneracy of eigenvalues everywhere in the phase. We offer an analytic, microscopic proof of this conjecture in the Read-Rezayi sequence of model states. We further identify a different ES that reflects the bulk quasihole spectrum and prove a bulk-edge correspondence in the ES. Finally, we show that the finite-size corrections of the ES of the Laughlin states reveal the fractionalization of the underlying quasiparticles.
The study of non-equilibrium behaviors of quantum matter is one of the research directions at the frontiers of condensed matter physics and, in general, of theoretical and experimental physics. Understanding the mechanisms which prevent many quantum particles from reaching thermal equilibrium can ultimately shed light on the foundations of statistical physics and provide insight into the anomalous long-time collective behaviors observed in experiments. In this thesis, we present new theoretical developments in this direction. In the first part of the thesis we discuss the non-equilibrium dynamics of the two-dimensional quantum Ising model, the prototypical model of magnetism. In particular, we investigate the dynamics of interfaces separating different spin domains. In some regimes, we can provide an exact description of their unitary time evolution, but we also discuss the robustness of our predictions away from the exactly-solvable limit. Moreover, we address the same problem in the presence of disorder and provide a hydrodynamic interpretation of our findings. In the second part of the thesis, we investigate the localization properties of disordered quantum systems. In particular, we develop a renormalization-group picture of the Anderson localization transition in generic geometries. This provides a new framework to understand the phase diagram of such systems and their critical behavior beyond the usual finite-size scaling analysis. We also provide further insights on this topic both by studying resonances and their connection with observables and by employing unsupervised non-parametric learning techniques to probe disordered systems in an experimentally accessible form. The results presented in this thesis constitute a step forward in understanding the non-equilibrium phases and regimes of quantum matter and they pave the way to interesting future developments.