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.
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.
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.
THESIS SUMMARYTEXT:This thesis is made of two parts. In the first one, the issue of entanglement in many body systems is addressed. The concept of entanglement and some of the recent progress on the study of entropy of entanglement in many body quantum systems are reviewed. Emphasis is placed on the scaling properties of entropy for one-dimensional models at quantum phase transitions. Then, we focus on the area-law scaling of the entanglement entropy. An explicit computation in arbitrary dimensions of the entanglement entropy of the ground state of a discretized scalar free field theory that shows the expected area law result is also presented. For this system, it is shown that area law scaling is a manifestation of a deeper reordering of the vacuum produced by majorization relations.To finish this first part, the issue of how simple can a quantum system be such as to give a highly entangled ground state is addressed. In particular, we propose a Hamiltonian of a XX model with a ground state whose entropy scales linearly with the size of the block. It provides a simple example of a one dimensional system of spin-1/2 particles with nearest neighbour interactions that violates area-law for the entanglement entropy.The second part of this thesis deals with the problem of simulating quantum mechanics for highly entangled systems. Two different approaches to this issue are considered. One consists of using ultra-cold atoms systems as quantum simulators. With this aim, some experimental techniques related to cold atoms that allow to simulate strongly correlated many body quantum systems are reviewed an explicit example of simulation is presented. In particular, we analyze how to achieve a Mott state of Laughlin wave functions in an optical lattice and study the consequences of considering anharmonic corrections to each single site potential expansion that were not taken into account until now.Finally, a different approach to simulate strongly correlated systems is considered: to use small quantum computers to simulate them. An explicit quantum algorithm that creates the Laughlin state for an arbitrary number of particles n in the case of falling fraction equal to one is presented. We further prove the optimality of the circuit using permutation theory arguments and we compute exactly how entanglement develops along the action of each gate. We also discuss its experimental feasibility decomposing the qudits and the gates in terms of qubits and two qubit-gates as well as the generalization to arbitrary falling fraction.KEYWORDS: Entanglement, Many body quantum systems, Quantum Information Condensed Matter, Cold atoms, Spin chains, Quantum simulator, Quantum computation.
This chapter provides a comprehensive introduction to the foundational concepts and advanced developments of mathematical perspectives on quantum information theory. Beginning with the distinctions between classical and quantum information, we explore key measures of quantum correlations such as entanglement entropy, mutual information, and entanglement of purification. Various entanglement quantifiers – including concurrence and negativity – are discussed in the context of bipartite and mixed states. The chapter also introduces quantum channels, noise models, and their physical realizations in systems ranging from superconducting qubits to black holes. We present key quantum algorithms, including Shor’s factoring algorithm and Grover’s search, along with their underlying principles such as the quantum Fourier transform. Bridging theory and practice, this chapter sets the stage for further exploration into quantum error correction, quantum thermodynamics, and the role of information in fundamental physics. The material is designed to be accessible to graduate students, researchers, and professionals across the disciplines of physics, computer science, and engineering.
One of the most important predictions of quantum mechanics is the existence of quantum entanglement. Since the inception of quantum theory, the study of entanglement has become ubiquitous in several branches of physics, from high energy to condensed matter physics and quantum information. To this avail, different measures of entanglement in quantum systems have been introduced. The most successful ones are the von Neumann and the Rényi entanglement entropies, which have been heavily studied both in lattice models and in quantum field theories (QFTs). A different, more complete characterization of entanglement is provided by the entanglement (or modular) Hamiltonian, defined as the logarithm of the reduced density matrix. One of the key property of this operator is that, under generic conditions, in 1+1-dimensional conformal field theories (CFTs) presents a local and few-body structure. Beyond just being of theoretical interest, this property has been recently used to efficiently construct the ground state of spin chains in quantum simulators. A recently introduced generalisation of the entanglement Hamiltonian is the negativity Hamiltonian, a non-Hermitian operator which is more useful for mixed states and tripartite entanglement. In this thesis we study these quantifications in many-body quantum systems and in QFTs. In the first part of the thesis, we focus on fermionic systems at equilibrium, and we obtain analytical expressions for the entanglement and the negativity Hamiltonians in the massless Dirac fermion QFT, both in presence of boundaries and at finite temperature. We compare these predictions in the continuum QFT with numerical calculations on the lattice, showing that a proper comparison requires a non-trivial continuum limit of the lattice expression, which we detail. We also investigate a lattice realisation of a non-unitary CFT, the bc-ghost theory, showing that the entanglement Hamiltonian presents an additional term with respect to the unitary case. In the second part, we study the entanglement entropies in an integrable QFT, the massless renormalisation group flow from the tricritical to the critical Ising CFT, demonstrating that the entropy is monotonically decreasing under the renormalisation group flow, as expected from the entropic c-theorem. In the third part we consider out-of-equilibrium problems. We study the melting of a domain wall configuration, computing analytically the entanglement Hamiltonian and showing that the insertion of a defect dramatically modifies the entanglement properties, changing the growth of the entropies from logarithmic to linear in time. Finally, we study generic quantum quenches in free fermionic theories, using the celebrated quasiparticle picture to obtain a prediction for the evolution of the entanglement Hamiltonian during the quench.
Discoveries in quantum materials, which are characterized by the strongly quantum-mechanical nature of electrons and atoms, have revealed exotic properties that arise from correlations. It is the promise of quantum materials for quantum information science superimposed with the potential of new computational quantum algorithms to discover new quantum materials that inspires this Review. We anticipate that quantum materials to be discovered and developed in the next years will transform the areas of quantum information processing including communication, storage, and computing. Simultaneously, efforts toward developing new quantum algorithmic approaches for quantum simulation and advanced calculation methods for many-body quantum systems enable major advances toward functional quantum materials and their deployment. The advent of quantum computing brings new possibilities for eliminating the exponential complexity that has stymied simulation of correlated quantum systems on high-performance classical computers. Here, we review new algorithms and computational approaches to predict and understand the behavior of correlated quantum matter. The strongly interdisciplinary nature of the topics covered necessitates a common language to integrate ideas from these fields. We aim to provide this common language while weaving together fields across electronic structure theory, quantum electrodynamics, algorithm design, and open quantum systems. Our Review is timely in presenting the state-of-the-art in the field toward algorithms with nonexponential complexity for correlated quantum matter with applications in grand-challenge problems. Looking to the future, at the intersection of quantum information science and algorithms for correlated quantum matter, we envision seminal advances in predicting many-body quantum states and describing excitonic quantum matter and large-scale entangled states, a better understanding of high-temperature superconductivity, and quantifying open quantum system dynamics.
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.