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arXiv:1205.3742

Entanglement in many-body quantum systems

J. I. Cirac

Published May 16, 2012• 1 citation
arXivHTMLPDF

Abstract

Short review on entanglement, as seen from a quantum information perspective, and some simple applications to many-body quantum systems. Special emphasis in area laws, cold atoms, and efficient descriptions using tensor network states.
DOI
10.48550/arxiv.1205.3742

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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.
We investigate the potential of tensor-network-based machine-learning methods to scale to large image and text datasets. We study how the mutual information between a subregion and its complement scales with the subsystem size L , similarly to approaches used in quantum many-body physics. Using various mutual information (MI) estimation methods, including an autoregressive network-based estimator, we find that simple image datasets exhibit area law scaling, suggesting efficient representation by two-dimensional tensor networks. More complex image datasets exceed the area law, indicating the need for generalized tensor network states or hybrid models. For Wikipedia text data, we observe power-law scaling I ( L ) ∝ L ν with ν ≈ 0.82 , approaching a volume law. This implies that one-dimensional tensor networks with area law entanglement may not efficiently capture the structure of text. We introduce two models to reproduce this scaling: a quantum-inspired random pair toy model and a linguistically motivated Markovian dependency tree model. In the latter model, matching the observed MI scaling allows us to infer the word-word correlation length distribution in text.