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.
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.
Entanglement is a principle concept in quantum information science and is central to an understanding of complex behaviour in strongly interacting, quantum many-body systems. This thesis applies the quantitative theory of entanglement to analyse composite quantum systems. We attempt to provide a simplified approach to quantum phase transitions in the context of dynamical bifurcations.We consider the entanglement in the ground state of quantum systems with well-defined semiclassical analogues. In particular, we are interested in semiclassical models that exhibit bifurcations of stationary points. We introduce previously unknown connections between the entanglement in the ground state of such quantum models and the bifurcations in the corresponding semiclassical limit. In the case of many body systems, the change in ground state entanglement at dynamical instabilities is related to the phenomenon of quantum phase transitions.By investigating the ground state entanglement in a number of representative quantum systems, a connection between dynamical bifurcations and quantum phase transitions facilitated by the entanglement is demonstrated. The interrelationships between the three phenomena have the potential to facilitate a more general understanding of quantum phase transitions.
Locality constrains the flow of information between different parts of many-body quantum systems. In quantum computers, this affects the ability to perform arbitrary interactions for quantum information processing tasks. A crucial challenge for scalable quantum architectures is thus to minimize the overheads due to locality constraints. Locality constraints also affect the way information and entanglement can be spread in many body quantum systems, and our ability to make predictions about such systems. One way to implement arbitrary interactions in locality-constrained architectures is by permuting qubits, also known as performing quantum routing. In the first part of this thesis, we investigate quantum routing strategies using emerging architectural capabilities beyond swap gates. We first investigate the use of mid-circuit measurements and classical feedback, which enable quantum teleportation. We prove upper bounds on the speedup afforded by the use of quantum teleportation for routing. We also design architectural connectivities which can exploit teleportation-based speedups. We then consider quantum routing by continuous-time Hamiltonian evolution, designing a routing algorithm that achieves a polynomial speedup in routing time over gate-based routing methods in architectures with a bottleneck, which is an intermediate region that constrains interactions between the rest of the device. We improve the lower bound on the time taken for routing in such systems, and develop a matching upper bound on the closely related task of generating and distributing entanglement. In the second part of this thesis, we investigate the behavior of finite-sized systems evolving by local Hamiltonians for a long duration. A major direction of research in condensed matter physics is the investigation of whether isolated systems thermalize (i.e., reach a state of thermal equilibrium). Various theoretical proposals such as information scrambling, entanglement between subsystems, and the eigenstate thermalization hypothesis have attempted to explain the conditions under which systems equilibrate. At the same time, notable counter-examples to thermalization have been found, such as systems that are integrable, exhibit many-body localization, or quantum many-body scars. We approach this question from a complexity-theoretic perspective. We show that the problem of deciding whether or not observables equilibrate to a givenvalue is PSPACE-complete. We also show that deciding whether observables relax to the thermal equilibrium value is contained in PSPACE, and is PSPACE-hard under quantum polynomial time reductions. These problems are thus intractable even for a quantum computer. Our results indicate that the proposed conditions leading to thermalization are either difficult to decide, or are not generic indicators of thermalization.
Quantum information theory (QIT) is an emerging field of physics which aims to develop new methods of dealing with information by harnessing the power of quantum mechanics. Besides its potential to revolutionize the techniques of information processing and communication, it also provides novel approaches to better comprehend the foundations of quantum mechanics. Among many important problems in QIT, manipulation and dynamical characterization of correlations present in quantum systems stand out due to their relevance for the practical applications of the theory. This thesis intends to explore such correlations of quantum and classical nature from various perspectives. In particular, our discussions involve the investigation of local transformations among a class of entangled states and the examination of correlation measures in some physical models. We first examine the classification of the flip (0-1) and exchange symmetric (FES) states under local quantum operations. We study the optimal local one-shot conversions of FES states to determine the entanglement transformations that relate multiqubit FES states with the maximum possible probability of success. Next, we investigate the exchange symmetry properties of certain symmetric states when the qubits evolve according to a dephasing model which is also invariant under swap operation. We find that there exist states which do not preserve the exchange symmetry with unit probability during the time evolution, leading to the spontaneous breaking of exchange symmetry. Later, we turn our attention to the dynamics of quantum and classical correlations for qubit-qutrit systems in independent and global dephasing environments. In these cases, we demonstrate several interesting phenomena such as the transition from classical to quantum decoherence. Lastly, we investigate the thermal quantum and total correlations in the one-dimensional anisotropic XY model in transverse field. We discuss the ability of different measures to estimate the critical point of the quantum phase transition at finite temperature. We also consider the relation between correlations and the factorized ground state in this model. Furthermore, we study the effect of temperature on long-range correlations.
Classical statistical mechanics relies on the ergodic hypothesis, which states that a system explores the entire allowed phase space during its time evolution. Pure quantum states are at odds with this concept as they remain in a single quantum state even if the system's parameters are suddenly altered. However, in a generic many-body system, coherent dynamics drive local thermalization of its subsystems. In this case, any local observable appears thermal and can be described by a statistical ensemble even though the full state remains pure. The key to understanding the phenomenon of quantum thermalization lies in the entanglement properties of the system: The presence of entanglement creates local entropy that validates the use of statistical physics.\nIn this thesis, we experimentally study the emergence of statistical mechanics in a quantum state and observe the fundamental role of quantum entanglement in facilitating this emergence. We perform microscopy on an evolving quantum system, and we see thermalization occur on a local scale while we measure that the full quantum state remains pure.\nThe only known robust exception from the paradigm of quantum thermalization is provided by disordered systems. In this case, the particle transport ceases through the system, preventing the formation of a sufficient amount of entanglement required for thermalization. However, if the many-body system is interacting, the state develops non-local quantum correlations that are inaccessible to any local observable. These correlations spread exponentially slowly in time throughout the entire system and are considered a hallmark feature of many-body-localized states.\nIn order to study the interplay between thermalization and localization in our system, we develop novel methods for high fidelity state preparation and readout. We achieved an exquisite isolation of our system from the environment, which allows us to monitor coherent many-body dynamics over multiple decades of time evolution.\nWe experimentally realize such a many-body localized system and observe the formation of non-local quantum correlations, while the particle transport is frozen in the system. Our work experimentally establishes many-body localization as a qualitatively distinct phenomenon from localization in non-interacting, disordered systems.
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.
The determination of structures that can emerge out of the interactions among the constituents of a quantum many-body system is a foundational task of condensed matter physics.One of the most advanced proposals within this paradigm is the holographic generation of spacetime in some strongly coupled chaotic systems with particular patterns of entanglement and quantum complexity, which is historically motivated by the holographic principle and by the AdS/CFT correspondence.As a part of this scenario, this thesis is dedicated to the analysis of some concepts that are relevant to such proposal, where they are separately applied to some lattice models.Concretely, the matter is divided into three main studies: first, the calculation of the time-dependent circuit complexity in the Ising model with a periodically driven transverse field, where we establish the effectiveness of this quantity for the detection of nonequilibrium quantum phase transitions.Our results provide hints for understanding how universal features out of equilibrium are captured by the complexity of quantum states.Second, the derivation of a bound on the maximal rate of entanglement entropy production for a class of one-dimensional quantum circuits with periodic dynamics.An example of a circuit that saturates the bound is composed by parallel SWAP gates acting on entangled pairs.Out of inequalities obeyed by the entropy, in addition to considerations on multipartite entanglement, we indicate that the effect of a chaotic dynamics cannot result in the increase on the rate of entanglement production.Third, the construction of a class of supersymmetric many-body systems using symmetric inverse semigroups.For particular toy models built out of this structure, we study some questions regarding supersymmetric phases, integrability, disorder, spreading of quantum information and many-body localization.Finally, besides those three works, we include an essay addressing the problem of emergent holographic spaces out of two-dimensional conformal field theories, where the mediation between the two parts is performed by means of a Riemannian structure defined in the Hilbert space of the field theory.