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arXiv:1010.4038

Mutual information and the structure of entanglement in quantum field theory

Brian G. Swingle

Published Oct 19, 2010• 48 citations
arXivHTMLPDF

Abstract

I study the mutual information between spatial subsystems in a variety of scale invariant quantum field theories. While it is derived from the bare entanglement entropy, the mutual information offers a more refined probe of the entanglement structure of quantum field theories because it remains finite in the continuum limit. I argue that the mutual information has certain universal singularities that are a manifestation of the idea of "entanglement per scale". Moreover, I propose a method, based on an ansatz for higher dimensional twist operators, to compute the entanglement entropy, Renyi entropy, and mutual information in a general quantum field theory. The relevance of these results to the search for renormalization group monotones, to holographic duality, and to entanglement based simulation methods for many body systems are all discussed.

DOI
10.48550/arxiv.1010.4038

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I study the mutual information between spatial subsystems in a variety of scale invariant quantum field theories. While it is derived from the bare entanglement entropy, the mutual information offers a more refined probe of the entanglement structure of quantum field theories because it remains finite in the continuum limit. I argue that the mutual information has certain universal singularities that are a manifestation of the idea of "entanglement per scale". Moreover, I propose a method, based on an ansatz for higher dimensional twist operators, to compute the entanglement entropy, Renyi entropy, and mutual information in a general quantum field theory. The relevance of these results to the search for renormalization group monotones, to holographic duality, and to entanglement based simulation methods for many body systems are all discussed.
Entanglement entropy is a proposal to quantify quantum entanglement of two disjoint regions in a pure system. It is a relatively new topic and is developing rapidly. The current main motives to study it are its applications to studying quantum gravity, thermalization and phase transitions in condensed matter systems. Mutual information is a related quantity and it can be used to measure the amount of information two disjoint regions share. The purpose of this thesis is to give an introduction to the general results of the field without considering specific systems. The two prominent approaches used are two-dimensional conformal field theory and the holographic entanglement entropy conjecture. The first approach is to calculate entanglement entropy using conformal field theory, the results of which are known to be exact although technically more difficult to derive and only available for 1+1 dimensional systems. Both static and dynamic systems will be discussed. The results are reproduced and generalized to higher dimensions using holography. As a more recent topic, the holographic approach is used to rederive the entanglement entropy of systems with a Fermi surface using AdS/Vaidya metric with Lifshitz scaling and hyperscaling violation. The final chapter of the thesis discusses mutual information in various static and dynamic cases considered in the previous chapters. For the first time, the evolution of mutual information is calculated in AdS/Vaidya metric with Lifshitz scaling and hyperscaling violation in the critical theta=d-1 case.
We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy S A = −Tr ρ A logρ A corresponding to the reduced density matrix ρ A of a subsystem A . For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c , we re-derive the result of Holzhey et al when A is a finite interval of length in an infinite system, and extend it to many other cases: finite systems, finite temperatures, and when A consists of an arbitrary number of disjoint intervals. For such a system away from its critical point, when the correlation length ξ is large but finite, we show that , where is the number of boundary points of A . These results are verified for a free massive field theory, which is also used to confirm a scaling ansatz for the case of finite size off-critical systems, and for integrable lattice models, such as the Ising and XXZ models, which are solvable by corner transfer matrix methods. Finally the free field results are extended to higher dimensions, and used to motivate a scaling form for the singular part of the entanglement entropy near a quantum phase transition.
We study a proper definition of Rényi mutual information (RMI) in quantum field theory as defined via the Petz Rényi relative entropy. Unlike the standard definition, the RMI we compute is a genuine measure of correlations between subsystems, as evidenced by its non-negativity and monotonicity under local operations. Furthermore, the RMI is UV finite and well defined in the continuum limit. We develop a replica path integral approach for the RMI in quantum field theories and evaluate it explicitly in 1+1D conformal field theory using twist fields. We prove that it bounds connected correlation functions and check our results against exact numerics in the massless free fermion theory.
Abstract We study different aspects of quantum field theory at finite density using methods from quantum information theory. For simplicity we focus on massive Dirac fermions with nonzero chemical potential, and work in 1 + 1 space-time dimensions. Using the entanglement entropy on an interval, we construct an entropic c-function that is finite. Unlike what happens in Lorentz-invariant theories, this c-function exhibits a strong violation of monotonicity; it also encodes the creation of long-range entanglement from the Fermi surface. Motivated by previous works on lattice models, we next calculate numerically the Renyi entropies and find Friedel-type oscillations; these are understood in terms of a defect operator product expansion. Furthermore, we consider the mutual information as a measure of correlation functions between different regions. Using a long-distance expansion previously developed by Cardy, we argue that the mutual information detects Fermi surface correlations already at leading order in the expansion. We also analyze the relative entropy and its Renyi generalizations in order to distinguish states with different charge and/or mass. In particular, we show that states in different superselection sectors give rise to a super-extensive behavior in the relative entropy. Finally, we discuss possible extensions to interacting theories, and argue for the relevance of some of these measures for probing non-Fermi liquids.
We carry out a systematic study of entanglement entropy in nonrelativistic conformal field theories via holographic techniques. After a discussion of recent results concerning Galilean conformal field theories, we deduce a novel expression for the entanglement entropy of (1+11+11+1)-dimensional Lifshitz field theories---this is done both at zero and finite temperature. Based on these results, we pose a conjecture for the anomaly coefficient of a Lifshitz field theory dual to new massive gravity. It is found that the Lifshitz entanglement entropy at finite temperature displays a striking similarity with that corresponding to a flat space cosmology in three dimensions. We claim that this structure is an inherent feature of the entanglement entropy for nonrelativistic conformal field theories. We finish by exploring the behavior of the mutual information for such theories.
This paper provides an in-depth analysis of entanglement entropy (EE) in quantum field theory (QFT), with a particular focus on its computation using the replica trick and its applications to both conformal and non-conformal systems. Beginning with an introduction to the basics of QFT, the study explains how entanglement entropy quantifies the quantum correlations between subsystems in a pure state, represented by the von Neumann entropy of the reduced density matrix. The replica trick is employed to derive the entanglement entropy, involving path integrals over n-sheeted Riemann surfaces. The paper demonstrates the method’s utility in 1+1-dimensional conformal field theory (CFT), where the central charge governs universal properties of entropy. It further extends the analysis to massive field theories, finite systems, and topological phases, exploring how deviations from conformal symmetry affect entanglement. The work also delves into the AdS/CFT correspondence, showcasing how holographic techniques facilitate entanglement entropy calculations in higher-dimensional systems via the Ryu-Takayanagi formula. In addition, the study investigates entanglement entropy in quantum lattice systems, considering the effects of spatial and thermal fluctuations. Numerical methods are used to compute the scaling of entanglement entropy with interval length for varying system masses and temperatures, highlighting the influence of these parameters on entropy behavior. Fractal lattice structures are explored to uncover unique entropy scaling laws and self-similar entanglement patterns. This study advances the under-standing of EE in diverse quantum systems and establishes a foundation for exploring its role in quantum criticality and fractal geometries. The paper also explores multiscale entanglement entropy (MSE), offering a comprehensive framework for future research in quantum critical phenomena and non-equilibrium systems, shedding light on the complex relationship between quantum entanglement and thermal effects. This work serves as a foundational reference for future theoretical and computational studies in quantum entanglement