Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Relative entropy close to the edge

Stefan Hollands

TL;DR

The paper proves a universal, model‑independent bound showing that the relative entropy between the vacuum’s reduced state on a region A and an excited state produced by a unitary localized near a boundary point p is largely insensitive to the global geometry of A, with deviations decaying sub‑exponentially as the localization scale ℓ shrinks. Grounded in Tomita–Takesaki modular theory and Araki’s relative entropy, the authors derive rigorous bounds that depend on the modular flow time and the energy content of the excitation, and they illustrate the results in QFT settings including touching regions in vacuum, free massless fermions in 1+1 dimensions, and large regions in thermal states. The work shows the dominant near‑boundary entanglement is governed by local geometry at p, not the global shape of A, and it opens avenues for interpreting entanglement changes via a local first‑law‑like relation. These results provide a robust, operator‑algebraic foundation for the locality‑driven structure of entanglement in QFT.

Abstract

We show that the relative entropy between the reduced density matrix of the vacuum state in some region $A$ and that of an excited state created by a unitary operator localized at a small distance $\ell$ of a boundary point $p$ is insensitive to the global shape of $A$, up to a small correction. This correction tends to zero as $\ell/R$ tends to zero, where $R$ is a measure of the curvature of $\partial A$ at $p$, but at a rate necessarily slower than $\sim \sqrt{\ell/R}$ (in any dimension). Our arguments are mathematically rigorous and only use model-independent, basic assumptions about quantum field theory such as locality and Poincare invariance.

Relative entropy close to the edge

TL;DR

The paper proves a universal, model‑independent bound showing that the relative entropy between the vacuum’s reduced state on a region A and an excited state produced by a unitary localized near a boundary point p is largely insensitive to the global geometry of A, with deviations decaying sub‑exponentially as the localization scale ℓ shrinks. Grounded in Tomita–Takesaki modular theory and Araki’s relative entropy, the authors derive rigorous bounds that depend on the modular flow time and the energy content of the excitation, and they illustrate the results in QFT settings including touching regions in vacuum, free massless fermions in 1+1 dimensions, and large regions in thermal states. The work shows the dominant near‑boundary entanglement is governed by local geometry at p, not the global shape of A, and it opens avenues for interpreting entanglement changes via a local first‑law‑like relation. These results provide a robust, operator‑algebraic foundation for the locality‑driven structure of entanglement in QFT.

Abstract

We show that the relative entropy between the reduced density matrix of the vacuum state in some region and that of an excited state created by a unitary operator localized at a small distance of a boundary point is insensitive to the global shape of , up to a small correction. This correction tends to zero as tends to zero, where is a measure of the curvature of at , but at a rate necessarily slower than (in any dimension). Our arguments are mathematically rigorous and only use model-independent, basic assumptions about quantum field theory such as locality and Poincare invariance.

Paper Structure

This paper contains 8 sections, 9 theorems, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Relative entropy between vacuum and an excited state
  3. Technical results
  4. Applications to quantum field theory
  5. Touching regions in vacuum
  6. Free massless fermions in $1+1$ dimensions
  7. Large regions in thermal states
  8. Conclusions

Key Result

Theorem 1

Let $U \in {\mathfrak M}_2$ be a unitary such that $\sigma_1^t(U) \in {\mathfrak M}_2$ for $|t| \le \tau$. Here, $H_-= -E_1^-\operatorname{ln} \Delta_1 \ge 0$ is the negative part of the modular Hamiltonian, where $E^-_1$ is the spectral projector of $\operatorname{ln} \Delta_1$ (see decomp) associated with the negative part of the spectrum.

Figures (3)

  • Figure 1: The two regions $A_1$ and $A_2$. The gray blob indicates the localization of $U$.
  • Figure 2: Causal diamond associated with $A$.
  • Figure 3: The regions $A_j, \ell B$.

Theorems & Definitions (17)

  • Theorem 1
  • Lemma 1
  • Remark 1
  • proof
  • Lemma 2
  • Remark 2
  • proof
  • Lemma 3
  • Theorem 2
  • Remark 3
  • ...and 7 more