# Entanglement at a Scale and Renormalization Monotones

## TL;DR

This paper introduces entanglement measures based on scaling and recovery maps in quantum field theory, showing they increase monotonically under renormalization group flow and unify various approaches to renormalization monotones.

## Contribution

It defines the entanglement of scaling and recovery as intrinsic measures in quantum field theory, extending the concept of renormalization monotones across different dimensions.

## Key findings

- Entanglement measures increase monotonically under RG flow.
- Vacuum state is a fixed point of scaling and recovery transformations.
- The framework applies to both relativistic and non-relativistic quantum field theories.

## Abstract

We study the information content of the reduced density matrix of a region in quantum field theory that cannot be recovered from its subregion density matrices. We reconstruct the density matrix from its subregions using two approaches: scaling maps and recovery maps. The vacuum of a scale-invariant field theory is the fixed point of both transformations. We define the entanglement of scaling and the entanglement of recovery as measures of entanglement that are intrinsic to the continuum limit. Both measures increase monotonically under the renormalization group flow. This provides a unifying information-theoretic structure underlying the different approaches to the renormalization monotones in various dimensions. Our analysis applies to non-relativistic quantum field theories as well the relativistic ones, however, in relativistic case, the entanglement of scaling can diverge.