This chapter reviews some of the basic concepts coming from the theory of entanglement, their applications to many-body systems, and some of the new theoretical methods that have come up in this context. Section 4.2 and 4.3 introduce entanglement for pure and mixed states, respectively. Two quantities are highlighted: the entropy of entanglement and the quantum mutual information. The first one measures the bipartite entanglement for pure states, whereas the second one measures correlations for both pure and mixed states. Section 4.4 concentrates on many-body states in thermal equilibrium that interact with short-range interactions in lattices. It shows that there is a common property of thermal states in lattice systems with short-range interactions, namely the area law. This fact can guide us in finding an efficient language to describe many-body quantum systems in which the number of parameters only scales polynomially with the number of sites. Under certain conditions, this is indeed possible in terms of so-called tensor network states. Section 4.5 considers how the area law leads to such descriptions and briefly reviews some of them while Section 4.6 concludes.
This chapter reviews some of the basic concepts coming from the theory of entanglement, their applications to many-body systems, and some of the new theoretical methods that have come up in this context. Section 4.2 and 4.3 introduce entanglement for pure and mixed states, respectively. Two quantities are highlighted: the entropy of entanglement and the quantum mutual information. The first one measures the bipartite entanglement for pure states, whereas the second one measures correlations for both pure and mixed states. Section 4.4 concentrates on many-body states in thermal equilibrium that interact with short-range interactions in lattices. It shows that there is a common property of thermal states in lattice systems with short-range interactions, namely the area law. This fact can guide us in finding an efficient language to describe many-body quantum systems in which the number of parameters only scales polynomially with the number of sites. Under certain conditions, this is indeed possible in terms of so-called tensor network states. Section 4.5 considers how the area law leads to such descriptions and briefly reviews some of them while Section 4.6 concludes.
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 deals with the novel characterisation of many-body physics using tools from quantum information theory. The first part focuses on the theory of entanglement, where the basic notions as bipartite versus multipartite entanglement, measures of entanglement, and the link between entangled states completely positive maps are discussed. The second part deals with the concept of area law, relating the von Neumann measure of the entropy of a subsystem with the size of the system for gapped, critical, and topological systems. The entanglement content in critical quantum systems is studied in reference to quantum phase transitions. Finally, the round states of many-body systems with a tensor network representation are described.
Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.
Quantum information theory has considerably helped in the understanding of quantum many-body systems. The role of quantum correlations and in particular, bipartite entanglement, has become crucial to characterise, classify and simulate quantum many body systems. Furthermore, the scaling of entanglement has inspired modifications to numerical techniques for the simulation of many-body systems leading to the, now established, area of tensor networks. However, the notions and methods brought by quantum information do not end with bipartite entanglement. There are other forms of correlations embedded in the ground, excited and thermal states of quantum many-body systems that also need to be explored and might be utilised as potential resources for quantum technologies. The aim of this work is to review the most recent developments regarding correlations in quantum many-body systems focussing on multipartite entanglement, quantum nonlocality, quantum discord, mutual information but also other non classical measures of correlations based on quantum coherence. Moreover, we also discuss applications of quantum metrology in quantum many-body systems.
We review some of the recent progress on the study of entropy of entanglement in many-body quantum systems. Emphasis is placed on the scaling properties of entropy for one-dimensional multi-partite models at quantum phase transitions and, more generally, on the concept of area law. We also briefly describe the relation between entanglement and the presence of impurities, the idea of particle entanglement, the evolution of entanglement along renormalization group trajectories, the dynamical evolution of entanglement and the fate of entanglement along a quantum computation.