Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Entanglement Entropy in CFT and Modular Nuclearity

Lorenzo Panebianco, Benedikt Wegener

TL;DR

The paper investigates entanglement in Algebraic Quantum Field Theory by leveraging the split property and canonical intermediate type I factors to define a finite canonical entanglement entropy $E_C(\omega)=S_\mathcal{F}(\omega)$ for chiral conformal nets. It proves finiteness of $E_C(\omega)$ for a broad class of nets (including the $U(1)$-current and $SU(n)$-loop group models) by constructing vacuum-preserving conditional expectations, and connects finiteness to modular nuclearity, deriving finite mutual information bounds under modular $p$-nuclearity with $0<p<1$. The work provides explicit upper bounds on entanglement measures, analyzes asymptotics and area-law-type behavior in 1+1D CFTs, and discusses asymptotic bounds in wedge-related QFT models, outlining several avenues for extending these results to wider model classes. Overall, the results strengthen the link between nuclearity properties and entanglement measures in AQFT and suggest concrete strategies for bounding entanglement in conformal and integrable QFTs.

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.

Entanglement Entropy in CFT and Modular Nuclearity

TL;DR

The paper investigates entanglement in Algebraic Quantum Field Theory by leveraging the split property and canonical intermediate type I factors to define a finite canonical entanglement entropy for chiral conformal nets. It proves finiteness of for a broad class of nets (including the -current and -loop group models) by constructing vacuum-preserving conditional expectations, and connects finiteness to modular nuclearity, deriving finite mutual information bounds under modular -nuclearity with . The work provides explicit upper bounds on entanglement measures, analyzes asymptotics and area-law-type behavior in 1+1D CFTs, and discusses asymptotic bounds in wedge-related QFT models, outlining several avenues for extending these results to wider model classes. Overall, the results strengthen the link between nuclearity properties and entanglement measures in AQFT and suggest concrete strategies for bounding entanglement in conformal and integrable QFTs.

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 and 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 -current model and the -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 -nuclearity condition for some . 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.

Paper Structure

This paper contains 10 sections, 16 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Mathematical background
  3. Von Neumann algebras inclusions
  4. Quantum entropy
  5. Entanglement entropy in chiral CFT
  6. Modular nuclearity and entanglement measures
  7. Modular nuclearity conditions
  8. Entanglement upper estimates
  9. Asymptotic in local QFT
  10. Outlook

Key Result

Theorem 1

Let $\mathcal{M}$ be a von Neumann algebra in standard form on $\mathcal{H}$. Let $\mathcal{N}$ be a von Neumann subalgebra of $\mathcal{M}$, $\omega$ a normal faithful state of $\mathcal{M}$ and $\xi$ the unique vector in the natural cone representing $\omega$. If there exists an $\omega$-preservin

Theorems & Definitions (43)

  • Theorem 1
  • proof
  • Remark 2
  • Corollary 3
  • proof
  • Definition 4
  • Definition 5
  • Lemma 6
  • proof
  • Corollary 7
  • ...and 33 more