This thesis is an attempt to apply the tools and techniques developed by quantum information science to gain Insight into strongly-interacting many-body quantum systems. The principle concept we make use of to do this is quantum entanglement. The thesis is divided into two parts. Part I is about quantum entanglement. The second part applies the techniques developed to understand quantum entanglement to complex quantum systems. In Part I we explore the rules governing the manipulation of quantum information. More specifically, we take as our premise the idea that quantum entanglement is the quantity responsible for the difference between classical and quantum information. We discuss the basic principles governing entanglement manipulation, focusing on the problem of quantifying quantum entanglement for quantum states. We prove that the probability that a certain class of quantum algorithms succeeds is an entanglement monotone for the initial state of the register used by these algorithms. We provide an easily-computable formula for a good mixed-state bipartite entanglement measure. We then use this formula to investigate a possible law of entanglement, namely, entanglement cannot be shared arbitrarily. In Part II we focus on using the concepts and tools of quantum information science to study complex quantum systems. The principal tool we use to do this is quantum entanglement. We begin with a definition of what we mean by a complex quantum system. This is followed with a study of the quantum entanglement present in a variety of complex quantum systems. We argue that quantum entanglement constitutes a signature of complex quantum behaviour. We conclude this thesis with a discussion of a well-known technique to understand complex quantum systems, namely, the density matrix renormalization group. We show that the success of this method is due to the way it treats the quantum entanglement present in a system.
This thesis is an attempt to apply the tools and techniques developed by quantum information science to gain Insight into strongly-interacting many-body quantum systems. The principle concept we make use of to do this is quantum entanglement. The thesis is divided into two parts. Part I is about quantum entanglement. The second part applies the techniques developed to understand quantum entanglement to complex quantum systems. In Part I we explore the rules governing the manipulation of quantum information. More specifically, we take as our premise the idea that quantum entanglement is the quantity responsible for the difference between classical and quantum information. We discuss the basic principles governing entanglement manipulation, focusing on the problem of quantifying quantum entanglement for quantum states. We prove that the probability that a certain class of quantum algorithms succeeds is an entanglement monotone for the initial state of the register used by these algorithms. We provide an easily-computable formula for a good mixed-state bipartite entanglement measure. We then use this formula to investigate a possible law of entanglement, namely, entanglement cannot be shared arbitrarily. In Part II we focus on using the concepts and tools of quantum information science to study complex quantum systems. The principal tool we use to do this is quantum entanglement. We begin with a definition of what we mean by a complex quantum system. This is followed with a study of the quantum entanglement present in a variety of complex quantum systems. We argue that quantum entanglement constitutes a signature of complex quantum behaviour. We conclude this thesis with a discussion of a well-known technique to understand complex quantum systems, namely, the density matrix renormalization group. We show that the success of this method is due to the way it treats the quantum entanglement present in a system.
This thesis addresses a number of problems within the emerging field of quantum information science. Quantum information science can be said to encompass the more-established disciplines of quantum computation and quantum information, as well as rather more recent attempts to apply concepts, tools and techniques from these disciplines to gain greater understanding of quantum systems in general. The role of entanglement — non-classical correlation — has been of particular interest to date. Part I contributes to this later goal. In particular, we establish a connection between the energy of a many-body quantum system and the idea of an entanglement witness from the theory of mixed-state entanglement. This connection allows mathematical results about entanglement witnesses to be translated into physical results about many-body quantum systems, specifically energy and temperature thresholds for entanglement. For the case of two qubits we are able to establish fairly detailed results about the behaviour of entanglement with temperature. We also study entanglement in systems of indistinguishable particles, where even the question of which quantum states should be regarded as entangled has been the subject of much controversy. We aim to clarify this issue by applying Wiseman and Vaccaro’s notion of entanglement of particles to a number of wellunderstood model systems. We discuss the advantages of the entanglement of particles approach compared with other methods in common use. Finally, we study the operational meaning of superselection rules in quantum physics, in particular the connection to the existence or not of an appropriate reference frame. We propose an experiment that aims to exhibit a coherent superposition of an atom and a molecule, apparently in violation of the commonly-accepted particle-number superselection rule. This result sheds light on the entanglement of particles approach to entanglement of indistinguishable particles. Part II returns to a fundamental question at the heart of quantum computation and quantum information, namely: how many quantum gates are required to perform a particular quantum computation? In other words, how efficiently can a quantum computer solve a particular computational problem? We establish a connection between this question and the field of Riemannian geometry. Intuitively, optimal quantum circuits correspond to “free-falling” along the shortest path between two points in a curved space. This opens up the possibility of using Riemannian geometry to study quantum computation, a possibility that was previously unknown. We provide explicit calculations of all the basic geometric quantities associated with the space, and give some preliminary results of applying geometric ideas to quantum computing. Finally, we explore more generally the connection between optimal control and quantum circuit complexity, of which the Riemannian metric described above can be viewed as a special case.
Quantum mechanics led us to reconsider the scope of physics and its building principles, such as the notions of realism and locality. More recently, quantum theory has changed in an equally dramatic manner our understanding of information processing and computation. On one hand, the fundamental properties of quantum systems can be harnessed to transmit, store, and manipulate information in a more efficient and secure way than possible in the realm of classical physics. On the other hand, the development of systematic procedures to manipulate systems of a large number of particles in the quantum regime, crucial to the implementation of quantum-based information processing, has triggered new possibilities in the exploration of quantum many-body physics and related areas. In this thesis, we present new results relevant to two important problems in quantum information science: the development of a theory of entanglement, intrinsically quantum correlations, and the exploration of the use of controlled quantum systems to the computation and simulation of quantum many-body phenomena. In the first part we introduce a new approach to the study of entanglement
The work that we present in this thesis tries to be at the crossover of quantum information science, quantum many-body physics, and quantum field theory. We use tools from these three fields to analyze problems that arise in the interdisciplinary intersection. More concretely, in Chapter 1 we consider the irreversibility of renormalization group flows from a quantum information perspective by using majorization theory and conformal field theory. In Chapter 2 we compute the entanglement of a single copy of a bipartite quantum system for a variety of models by using techniques from conformal field theory and Toeplitz matrices. The entanglement entropy of the so-called Lipkin-Meshkov-Glick model is computed in Chapter 3, showing analogies with that of (1+1)-dimensional quantum systems. In Chapter 4 we apply the ideas of scaling of quantum correlations in quantum phase transitions to the study of quantum algorithms, focusing on Shor's factorization algorithm and quantum algorithms by adiabatic evolution solving an NP-complete and the searching problems. Also, in Chapter 5 we use techniques originally inspired by condensed-matter physics to develop classical simulations, using the so-called matrix product states, of an adiabatic quantum algorithm. Finally, in Chapter 6 we consider the behavior of some families of quantum algorithms from the perspective of majorization theory. The structure within each Chapter is such that the last section always summarizes the basic results. Some general conclusions and possible future directions are briefly discussed in Chapter 7. Appendix A, Appendix B and Appendix C respectively deal with some basic notions on majorization theory, conformal field theory, and classical complexity theory.
Recently developed quantum algorithms suggest that quantum computers can solve certain problems and perform certain tasks more efficiently than conventional computers. Among other reasons, this is due to the possibility of creating non-classical correlations, or quantum entanglement, which is a phenomena hard or impossible to reproduce by classical-information methods. In this thesis I first investigate the simulation of quantum systems on a quantum computer constructed of two-level quantum elements or qubits. For this purpose, I present algebra mappings that allow one to obtain physical properties and compute correlation functions of fermionic, anyonic, and bosonic systems with such a computer. The results obtained show that the complexity of preparing a quantum state which contains the desired information for the computation is crucial. Second, I present a wide class of quantum computations, which could involve entangled states, that can be simulated with the same efficiency on both types of computers. The notion of generalized quantum entanglement then naturally emerges. This generalization of entanglement is based on the idea that entanglement is an observer-dependent concept, that is, relative to a set of preferred observables.
Entanglement and circuit complexity are fundamental concepts in quantum information theory which turned out to have considerable applications in various contexts ranging from many-body systems and condensed matter theory to quantum gravity and black hole physics. In order to find connections between these aspects, an insightful approach consists in comparing the complexity with some entanglement measures of a given bipartition, as for instance the Von Neumann entanglement entropy. Since, for a system in a given quantum state, the bipartite entanglement can be determined once a spatial subsystem and its complement are chosen, this comparison has to be made considering the complexity of reduced density matrices. This motivates the development of techniques for determining the circuit complexity of mixed states. In this thesis, focussing on free bosonic systems, we study the circuit complexity of mixed Gaussian states by employing the Fisher information geometry for the covariance matrices. This approach allows computing the complexity between reduced density matrices (subsystem complexity), as well as the complexity between thermal states. It also leads to a precise prescription for computing the spectrum and the basis complexity and the temporal evolution of the subsystem complexity after global and local quantum quenches. The study of the subsystem complexity of a given bipartition and a complete characterisation of its entanglement requires the detailed knowledge of the corresponding reduced density matrix, or, equivalently, of the entanglement Hamiltonian. A part of this thesis is devoted to determining the entanglement Hamiltonians in free theories. At equilibrium, we consider gapless systems, with the aim of recovering the underlying CFT results, and gapped lattice models. The evolution of entanglement Hamiltonians in quantum chains after a global quench is also studied. In addition, other entanglement quantifiers, as the entanglement spectrum and the contour for the entanglement entropies, are discussed in some of these settings. Finally, motivated by theoretical and experimental advances, we also address the issue of how the entanglement splits into the different charge sectors of a system endowed with a U(1) symmetry. We focus in particular on massive theories, either continuous or on the lattice, and we try to understand whether the entanglement equipartition, which is known to characterise gapless systems, survives away from criticality.
This thesis investigates questions related with the quantification, creation and application of entanglement. We first show how the notion of entanglement is modified by the presence of superselection rules. Next, we develop protocols for the optimal generation of squeezing and entanglement from noisy operations. The second part of the thesis applies quantum information methods to quantum many-body problems. We start by investigating the relation of Hamiltonian and correlations for lattices of harmonic oscillators. Next, we turn towards the description of many-body systems in terms of Matrix Product States (MPS) and Projected Entangled Pair States (PEPS). We clarify the relation between entropy scaling laws and the approximability by MPS, and determine the computational complexity of creating and simulating PEPS. Finally, we define Gaussian MPS as a generalization of MPS to bosonic systems.