Symmetry resolved relative entropies and distances in conformal field theory

TL;DR

This paper develops a systematic method to compute symmetry-resolved relative entropies and distances in 1+1D conformal field theories, revealing entanglement equipartition across symmetry sectors and providing analytic formulas with numerical validation.

Contribution

It introduces a novel approach to calculate symmetry-resolved entanglement measures and demonstrates entanglement equipartition in CFTs, extending understanding of quantum correlations in symmetry sectors.

Findings

Relative entropies and distances are identical across symmetry sectors.

Analytic expressions for charged moments are derived and validated.

Entanglement equipartition holds for the measures studied.

Abstract

We develop a systematic approach to compute the subsystem trace distances and relative entropies for subsystem reduced density matrices associated to excited states in different symmetry sectors of a 1+1 dimensional conformal field theory having an internal U(1) symmetry. We provide analytic expressions for the charged moments corresponding to the resolution of both relative entropies and distances for general integer $n$. For the relative entropies, these formulas are manageable and the analytic continuation to $n=1$ can be worked out in most of the cases. Conversely, for the distances the corresponding charged moments become soon untreatable as $n$ increases. A remarkable result is that relative entropies and distances are the same for all symmetry sectors, i.e. they satisfy entanglement equipartition, like the entropies. Moreover, we exploit the OPE expansion of composite twist fields, to provide very general results when the subsystem is much smaller than the total system. We focus on the massless compact boson and our results are tested against exact numerical calculations in the XX spin chain.