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A covariant regulator for entanglement entropy: proofs of the Bekenstein bound and QNEC

Jonah Kudler-Flam, Samuel Leutheusser, Adel A. Rahman, Gautam Satishchandran, Antony J. Speranza

Abstract

While von Neumann entropies for subregions in quantum field theory universally contain ultraviolet divergences, differences between von Neumann entropies are finite and well-defined in many physically relevant scenarios. We demonstrate that such a notion of entropy differences can be rigorously defined in quantum field theory in a general curved spacetime by introducing a novel, covariant regulator for the entropy based on the modular crossed product. This regulator associates a type II von Neumann algebra to each spacetime subregion, resulting in well-defined renormalized entropies. This prescription reproduces formulas for entropy differences that coincide with heuristic formulas widely used in the literature, and we prove that it satisfies desirable properties such as unitary invariance and concavity. As an application, we provide proofs of the Bekenstein bound and the quantum null energy condition, formulated directly in terms of vacuum-subtracted von Neumann entropies.

A covariant regulator for entanglement entropy: proofs of the Bekenstein bound and QNEC

Abstract

While von Neumann entropies for subregions in quantum field theory universally contain ultraviolet divergences, differences between von Neumann entropies are finite and well-defined in many physically relevant scenarios. We demonstrate that such a notion of entropy differences can be rigorously defined in quantum field theory in a general curved spacetime by introducing a novel, covariant regulator for the entropy based on the modular crossed product. This regulator associates a type II von Neumann algebra to each spacetime subregion, resulting in well-defined renormalized entropies. This prescription reproduces formulas for entropy differences that coincide with heuristic formulas widely used in the literature, and we prove that it satisfies desirable properties such as unitary invariance and concavity. As an application, we provide proofs of the Bekenstein bound and the quantum null energy condition, formulated directly in terms of vacuum-subtracted von Neumann entropies.
Paper Structure (4 sections, 5 theorems, 87 equations, 2 figures)

This paper contains 4 sections, 5 theorems, 87 equations, 2 figures.

Table of Contents

  1. Comments on Hermitian forms and domains
  2. Details on regulated entropy
  3. Proof of the Bekenstein Bound
  4. Proof of the Quantum Null Energy Condition

Key Result

Lemma 2.1

The image of the operator $-\log W+\log(g(\hat{X})\Delta)$ under the UCP map $\langle f|\cdot|f\rangle$ is given by where and $R(\Delta,h)$ is a positive operator defined by

Figures (2)

  • Figure 1: The entanglement entropy of a causally complete spacetime subregion $\mathcal{R}$ is divergent due to high-energy modes (red) across the entangling surface. Here $\Sigma_{\mathcal{R}}$ is a Cauchy surface for $\mathcal{R}$ and its boundary is the entangling surface.
  • Figure 2: The null plane (purple) in Minkowski spacetime is partitioned by the wiggly cut $V(y)$. This defines the wiggly Rindler wedge as the region to the right of the cut, with the green plane forming the past null boundary.

Theorems & Definitions (10)

  • Lemma 2.1: Faulkner2023
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 3.1: Bekenstein Bound
  • proof
  • Theorem 4.1: Quantum Null Energy Condition
  • proof