Entanglement in Many Body Quantum Systems

Abstract

THESIS SUMMARY<br/><br/>TEXT:<br/><br/>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. <br/><br/>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.<br/><br/>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.<br/><br/>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.<br/><br/>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.<br/><br/>KEYWORDS: Entanglement, Many body quantum systems, Quantum Information Condensed Matter, Cold atoms, Spin chains, Quantum simulator, Quantum computation.

<i>"Entrellaçament quàntic en sistemes de molts cossos"<br/><br/>TEXT:<br/>Aquesta tesi està composada per dues parts. En la primera, adrecem la qüestió de l'entrellaçament quàntic en els sistemes de molts cossos. Així, introduïm primer el concepte d'entrellaçament i revisem els progressos recents sobre aquest camp. A continuació, ens centrem la llei d'àrea per l'entropia d'entrellaçament i presentem un càlcul explícit d'aquesta entropia per a l'estat fonamental d'un camp escalar no interactuant obtenint la llei d'àrea esperada. Finalment, acabem aquesta part presentant un sistema molt senzill 1-dimensional que tot i tenir interaccions locals mostra una llei de volum per l'entropia.<br/><br/>En la segona part de la tesi tractem el problema de la simulació de sistemes quàntics altament entrellaçats. Considerem dos possibles vies per tractar aquest problema. Una d'elles consisteix en la utilització d'àtoms ultra-freds com a simuladors quàntics. En particular, analitzem un mètode per obtenir un estat producte de funcions d'ona de Laughlin en un xarxa òptica i estudiem les conseqüències de considerar la correcció anharmònica de l'expansió del potencial a cada pou de la xarxa. Finalment, considerem una altra aproximació a la simulació de sistemes fortament correlacionats: utilitzar petits ordinadors quàntics per a simular-los. Per il.lustrar aquest tipus de simulació, presentem un algoritme quàntic que crea un estat de Laughlin per un nombre arbitrari de partícules i en el cas de fracció d'ocupació 1. </i>