In the present thesis we investigate how the quantum entanglement of multicomponent systems evolves under realistic conditions. More specifically, we focus on open quantum systems coupled to the (uncontrolled) degrees of freedom of an environment. We identify key quantities that describe the entanglement dynamics, and provide efficient tools for its calculation. For quantum systems of high dimension, entanglement dynamics can be characterized with high precision. In the first part of this work, we derive evolution equations for entanglement. These formulas determine the entanglement after a given time in terms of a product of two distinct quantities: the initial amount of entanglement and a factor that merely contains the parameters that characterize the dynamics. The latter is given by the entanglement evolution of an initially maximally entangled state. A maximally entangled state thus benchmarks the dynamics, and hence allows for the immediate calculation or - under more general conditions - estimation of the change in entanglement. Thereafter, a statistical analysis supports that the derived (in-)equalities describe the entanglement dynamics of the majority of weakly mixed and thus experimentally highly relevant states with high precision. The second part of this work approaches entanglement dynamics from a topological perspective. This allows for a quantitative description with a minimum amount of assumptions about Hilbert space (sub-)structure and environment coupling. In particular, we investigate the limit of increasing system size and density of states, i.e. the macroscopic limit. In this limit, a universal behaviour of entanglement emerges following a "reference trajectory", similar to the central role of the entanglement dynamics of a maximally entangled state found in the first part of the present work.
In the present thesis we investigate how the quantum entanglement of multicomponent systems evolves under realistic conditions. More specifically, we focus on open quantum systems coupled to the (uncontrolled) degrees of freedom of an environment. We identify key quantities that describe the entanglement dynamics, and provide efficient tools for its calculation. For quantum systems of high dimension, entanglement dynamics can be characterized with high precision. In the first part of this work, we derive evolution equations for entanglement. These formulas determine the entanglement after a given time in terms of a product of two distinct quantities: the initial amount of entanglement and a factor that merely contains the parameters that characterize the dynamics. The latter is given by the entanglement evolution of an initially maximally entangled state. A maximally entangled state thus benchmarks the dynamics, and hence allows for the immediate calculation or - under more general conditions - estimation of the change in entanglement. Thereafter, a statistical analysis supports that the derived (in-)equalities describe the entanglement dynamics of the majority of weakly mixed and thus experimentally highly relevant states with high precision. The second part of this work approaches entanglement dynamics from a topological perspective. This allows for a quantitative description with a minimum amount of assumptions about Hilbert space (sub-)structure and environment coupling. In particular, we investigate the limit of increasing system size and density of states, i.e. the macroscopic limit. In this limit, a universal behaviour of entanglement emerges following a "reference trajectory", similar to the central role of the entanglement dynamics of a maximally entangled state found in the first part of the present work.
This thesis presents studies performed on open quantum systems, that is, quantum systems interacting with their surrounding environment. Such systems are important not only in understanding the quantum-to-classical transition but also for the practical implementation of modern quantum technologies. In studies of open quantum systems performed to date, a very common assumption is that the system and the environment are in separated initial states to begin with. One primary objective of this thesis is to critically analyse this assumption. We follow two different approaches to investigate the dynamics. First, we solve an exactly solvable spin-spin model where a central spin system interacts with a collection of quantum spins. We analyse exactly the central spin dynamics, starting from both initially correlated and uncorrelated SE states, and look at the dynamical differences due to the different starting states. Second, we consider an arbitrary system interacting with an arbitrary environment and derive a master equation that describes the system dynamics and incorporates the effect of the initial SE correlations. This effect of initial correlations is captured by an extra term appearing in the master equation. The master equation is subsequently applied to the paradigmatic SE models such as the spin-boson model and the spin-spin model. We demonstrate that the role played by initial correlations can be noticeable even if the SE coupling strength is kept smaller. The next part of the thesis deals with estimating the parameters characterizing the environment of a quantum system where we show that the use of two two-level systems can greatly enhance the estimation of the environment parameters.
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)].
We show that the entanglement evolution of an open quantum system is the same for the vast majority of initial pure states, in the limit of large Hilbert space dimensions. The analytical results are illustrated by a numerical study of the evolution of negativity for many-qubit systems undergoing a local decoherence process. In the limit of many subsystems, the arising mixed state entanglement between one subsystem and the rest grows more robust, whereas for two equally sized partitions it becomes more fragile.
In contrast to classical many-particle systems for which the number of degrees of freedom grows linearly with the system size, it grows exponentially for quantum many-particle systems. On the one hand, the resulting complexity of the quantum systems has a high potential for technological advances if it can be exploited directly. For example, in recent years, the fundamentals for the realization of quantum computers and their possible applications have been investigated intensely. On the other hand, the high complexity, especially in strongly correlated regimes, often makes the analytical or numerical analysis of underlying physical phenomena very difficult. This is why some systems of high technological relevance, like the high-temperature superconductors, are still unsatisfactorily understood. A main topic of this thesis is the development of efficient numerical methods for the simulation of strongly correlated quantum lattice models. For one-dimensional systems, the density-matrix renormalization-group (DMRG) is such a very successful method. The physical states of interest are approximated within a certain class of ansatz states. These ansatz states are designed in a way that the number of degrees of freedom are prevented from growing exponentially. With the DMRG method, ground states of interacting one-dimensional systems can often be determined with very high precision. This observation indicates already that ground states actually do not exhaust the available exponential number of degrees of freedom but can be described efficiently with much fewer parameters. The ansatz states used in DMRG are the so-called matrix product states. A very useful criterion for the construction of corresponding appropriate ansatz states for higher-dimensional systems is the quantum nonlocality as for example quantified by the entanglement entropy. The ansatz states should be able to reproduce the scaling properties of the quantum nonlocality in the physical states of interest with as few parameters as possible. The first part of the thesis, therefore, provides analytical and numerical analysis of the scaling of quantum nonlocality with the system size or time in different, physically relevant scenarios. For example, the scaling of Renyi entropies and their dependence on boundary conditions is derived within the 1+1-dimensional conformal field theory. Conjectures and analytical indications concerning the properties of entanglement entropy in critical fermionic and bosonic systems are confirmed numerically with high precision. For integrable models in the thermodynamic limit, general preconditions are derived under which subsystems converge to steady states. These steady states are non-thermal and retain information about the initial state. It is shown that the entanglement entropy in such steady states is extensive. For short times, the entanglement entropy grows typically linearly with time, causing an exponential increase in computation costs for the DMRG method. Consequently, time-dependent DMRG simulations usually have no direct access to the long time regime. The second part of the thesis focuses on the development and improvement of the above-mentioned numerical techniques. The time-dependent DMRG is complemented with an extrapolation technique for the evaluated observables. In this way, the problem of the entropy increase can be circumvented, allowing for a precise determination of spectral functions. The method is demonstrated using the example of the Heisenberg antiferromagnet and results are compared to Bethe-Ansatz data for T=0 and quantum Monte Carlo data for T>0. For the simulation of higher-dimensional systems, projected-entangled pair-states (PEPS) and the multiscale entanglement renormalization ansatz (MERA) turn out to be appropriate wavefunction classes. Initially, they could only be used for spin systems. Within this thesis, variants of those state classes are being suggested for the fermionic case. It is shown how the known algorithms for spin systems can be translated into corresponding algorithms for the fermionic systems. The occurring computation costs increase only by a marginal overhead. First numerical benchmarking tests proceeded successfully. Thus, new techniques are available which can for example be used to examine the two-dimensional Hubbard model. This model is a candidate for the description of high-temperature superconductivity.
This thesis, explores the quantum entanglement and evolution through both a geometric and dynamical perspective. The first part focuses on classical phase space and its central role in Hamiltonian mechanics, emphasizing the importance of symplectic structures in describing mechanical states. The study highlights the formal analogy between classical phase space and the Hilbert space used in quantum mechanics. The second part is devoted to the geometric description of quantum states through the projective structure of Hilbert space. Emphasis is placed on the geometric interpretation of quantum evolution, particularly via the Fubini-Study metric, associated symplectic structures, and the geometric phase acquired during unitary evolutions. The final two parts are dedicated to the study of spin systems (both two-body and many-body) under different interaction models (XXZ Heisenberg and all-range Ising). Both the dynamical aspects (evolution speed, entanglement, and the quantum brachistochrone problem) and the geometric and topological structures of the corresponding quantum states are analyzed.
The thesis presents research related to the dynamics of quantum systems, both isolated and in the presence of interactions with their environment. Generally, I employ matrix product state (MPS) techniques to explore quantum dynamics in open and closed systems. In the first part I present a study of quantum chaos and how mapping to a MPS variational manifold allow the use of techniques developed in the study of classical many-body systems. Using code developed for this project the Lyapunov spectrum is extracted to give an alternative perspective on eigenstate thermalization, pre-thermalization and integrability. In the second part, I present a novel combination of MPS methods with a Langevin description of the open system. I use this to show how coupling to the environment restricts the growth of entanglement. The consequences of this are relevant for simulations of open quantum systems and their use in in quantum technologies. Finally I discuss applications of these ideas to quantum search. I consider adiabatic and quantum walk algorithms for optimal scaling quantum search algorithms, and hybridizations between the two. The robustness of the different underlying physical mechanisms is investigated in a simple infinite-temperature model, and in a low-temperature limit using the MPS Langevin equation.