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arXiv:2108.09074

Entanglement Entropy in CFT and Modular Nuclearity

Lorenzo Panebianco, Benedikt Wegener

Published Aug 20, 2021• 0 citations
arXivHTMLPDF

Abstract

In the framework of Algebraic Quantum Field Theory, several operator algebraic notions of entanglement entropy can be associated with any pair of causally disjoint spacetime regions $\mathcal{S}_A$ and $\mathcal{S}_B$ with positive relative distance. Among them, the canonical entanglement entropy is defined as the von Neumann entropy of a canonical intermediate type I factor. In this work, we show that the canonical entanglement entropy of the vacuum state is finite for a broad class of conformal nets including the $U(1)$-current model and the $SU(n)$-loop group models. Since previous studies suggest that this finiteness property is related to nuclearity properties of the system, we show that the mutual information is finite in any local QFT satisfying a modular $p$-nuclearity condition for some $0 < p < 1$. A similar finiteness result is established for another notion of entanglement entropy introduced in this paper. We conclude with remarks for future work in this direction.

DOI
10.48550/arxiv.2108.09074

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•Mar 23, 2017•2 citations
In the framework of Algebraic Quantum Field Theory, several operator algebraic notions of entanglement entropy can be associated with any pair of causally disjoint spacetime regions SA\mathcal{S}_ASA​ and SB\mathcal{S}_BSB​ with positive relative distance. Among them, the canonical entanglement entropy is defined as the von Neumann entropy of a canonical intermediate type I factor. In this work, we show that the canonical entanglement entropy of the vacuum state is finite for a broad class of conformal nets including the U(1)U(1)U(1)-current model and the SU(n)SU(n)SU(n)-loop group models. Since previous studies suggest that this finiteness property is related to nuclearity properties of the system, we show that the mutual information is finite in any local QFT satisfying a modular ppp-nuclearity condition for some 0<p<10 < p < 10<p<1. A similar finiteness result is established for another notion of entanglement entropy introduced in this paper. We conclude with remarks for future work in this direction.
In the framework of Algebraic Quantum Field Theory, several operator algebraic notions of entanglement entropy can be associated to a couple of causally disjoint and distant spacetime regions SA\mathcal{S}_ASA​ and SB\mathcal{S}_BSB​. In this work, we show that the mutual information is finite in any local QFT verifying a modular ppp-nuclearity condition for some 0<p<10 < p <10<p<1. A similar result is proved for another recently studied entanglement measure. Furthermore, if we assume conformal covariance then by comparison with other entanglement measures we can state that the mutual information satisfies lower bounds of area law type when the distance between SA\mathcal{S}_ASA​ and SB\mathcal{S}_BSB​ approaches to zero. As an application, in 1+11+11+1-dimensional integrable models with factorizing S-matrices, we study the asymptotic behaviour of different entanglement measures as the distance between two causally disjoint wedges diverges.
In this thesis, we study aspects of entanglement theory of quantum field theories from an algebraic point of view. The main motivation is to gain insights about the general structure of the entanglement in QFT, towards a definition of an entropic version of QFT. In the opposite direction, we are also interested in exploring any consequence of the entanglement in algebraic QFT. This may help us to reveal unknown features of QFT, with the final aim of finding a dynamical principle which allows us to construct non-trivial and rigorous models of QFT. The algebraic approach is the natural framework to define and study entanglement in QFT, and hence, to pose the above inquiries. After a self-contained review of algebraic QFT and quantum information theory in operator algebras, we focus on our results. We compute, in a mathematically rigorous way, exact solutions of entanglement measures and modular Hamiltonians for specific QFT models, using algebraic tools from modular theory of von Neumann algebras. These calculations show explicitly non-local features of modular Hamiltonians and help us to solve ambiguities that arise in other non-rigorous computations. We also study aspects of entanglement entropy in theories having superselection sectors coming from global symmetries. We follow the algebraic perspective of Doplicher, Haag, and Roberts. In this way, we find an entropic order parameter that "measures" the size of the symmetry group, which is made out of a difference of two mutual informations. Moreover, we identify the main operators that take account of such a difference, and we obtain a new quantum information quantity, the entropic certainty relation, involving algebras containing such operators. This certainty relation keeps an intrinsic connection with subfactor theory of von Neumann algebras.
Abstract Given a QFT net A{{\mathcal {A}}}A A of local von Neumann algebras A(O){{\mathcal {A}}}(O)A(O) A ( O ) , we consider the von Neumann entropy SA(O,O~)S_{{\mathcal {A}}}(O, {\widetilde{O}})SA​(O,O) S A ( O , O ~ ) of the restriction of the vacuum state to the canonical intermediate type I factor for the inclusion of von Neumann algebras A(O)⊂A(O~){{\mathcal {A}}}(O)\subset {{\mathcal {A}}}({\widetilde{O}})A(O)⊂A(O) A ( O ) ⊂ A ( O ~ ) (split property). This canonical entanglement entropy SA(O,O~)S_{{\mathcal {A}}}(O, {\widetilde{O}})SA​(O,O) S A ( O , O ~ ) is finite for the chiral conformal net on the circle generated by finitely many free Fermions (here double cones are intervals). The finiteness property is derived by an explicit formula of entropy and an observation that the operators in the definition are closely related to Hankel operators. In this paper we give further analysis of this entropy using a variety of techniques that have been developed in different context, and in particular we show that there is an upper bound given by a positive constant multiply by ∣ln⁡η∣|\ln \eta |∣lnη∣ | ln η | , where η\eta η η is the cross ratio of the underlying system, when η→0\eta \rightarrow 0η→0 η → 0 .
An early result of algebraic quantum field theory is that the algebra of any subregion in a QFT is a von Neumann factor of type III _{1} 1 , in which entropy cannot be well-defined because such algebras do not admit a trace or density states. However, associated to the algebra is a modular group of automorphisms characterizing the local dynamics of degrees of freedom in the region, and the crossed product of the algebra with its modular group yields a type II _∞ ∞ factor, in which traces and hence von Neumann entropy can be well-defined. In this work, we generalize recent constructions of the crossed product algebra for the TFD to, in principle, arbitrary spacetime regions in arbitrary QFTs, formally paving the way to the study of entanglement entropy without UV divergences. In contrast to previous works, we emphasize that this construction is independent of gravity. In this sense, the crossed product construction represents a refinement of Haag’s assignment of nets of observable algebras to spacetime regions by providing a natural construction of a type II factor. We present several concrete examples: a QFT in Rindler space, a CFT in an open ball of Minkowski space, and arbitrary boundary subregions in AdS/CFT. In the holographic setting, we provide a novel argument for why the bulk dual must be the entanglement wedge, and discuss the distinction arising from boundary modular flow between causal and entanglement wedges for excited states and disjoint regions.
Quantum Field Theory (QFT) represents a vast generalization of Quantum Mechanics (QM), as it deals with systems that have an infinite number of degrees of freedom. The Stone-von Neumann theorem, which establishes the equivalence of irreducible representations of the canonical commutation relations (CCR) in QM, does not extend to QFT. Consequently, QFT admits multiple inequivalent irreducible representations, leading to a much richer algebraic structure. This essay aims to explore the physics of QFT from the operator algebra perspective, particularly focusing on entanglement entropy. We discuss the role of von Neumann algebras of different types in QFT, describe the local operator algebra approach to QFT, and explain how entanglement entropy can be defined in terms of the algebra of observables. Additionally, we explore the benefits of this approach in concrete applications, specifically in quantum field theory on curved spacetime.
We consider the problem of defining the entanglement entropy for chiral nets in the framework of algebraic quantum field theory.Considering a Möbius covariant local net with the split property, we give a sensible definition for the entropy H E I of a state restricted to a local algebra of an open connected non-dense interval I ⊂ S 1 , with a given "conformal energy cutoff E".Considering the vacuum state restricted to any such interval, we prove that the latter is finite, and we give some upper bound estimates in terms of the dimensions of eigenspaces of the conformal Hamiltonian.This thesis is based on a joint work with Yoh Tanimoto.