Real-space entanglement of quantum fields

TL;DR

This paper introduces an analytical method to compute entanglement measures between spatial regions of quantum fields, providing explicit formulas and confirming known decay behaviors with new exact results.

Contribution

It develops a novel analytical framework for entanglement in quantum fields, enabling explicit calculations of entropy, mutual information, and discord based on Fourier spectra.

Findings

Derived exact formulas for entanglement entropy, mutual information, and discord.

Confirmed mutual information decays as the fourth power of the ratio of region size to distance.

Found quantum discord vanishes unless coarse-graining involves smeared spheres.

Abstract

We introduce a new method permitting the analytical determination of entanglement entropy (and related quantities) between configurations of a quantum field, which is either free or in interaction with a classical source, at two distinct spatial locations. We show how such a setup can be described by a bipartite, continuous Gaussian system. This allows us to derive explicit and exact formulas for the entanglement entropy, the mutual information and the quantum discord, solely in terms of the Fourier-space power spectra of the field. This contrasts with previous studies, which mostly rely on numerical considerations. As an illustration, we apply our formalism to massless fields in flat space, where exact expressions are derived that only involve the ratio between the size of the regions over which the field is coarse-grained, and the distance between these regions. In particular, we recover the well-known fact that mutual information decays as the fourth power of this ratio at large distances, as previously observed in numerical works. Our method leads to the first analytical derivation of this result, and to an exact formula that also applies to arbitrary distances. Finally, we determine the quantum discord and find that it identically vanishes (unless coarse-graining is performed over smeared spheres, in which case it obeys the same suppression at large distance as mutual information).