Symmetry-resolved Page curves

TL;DR

This paper introduces symmetry-resolved Page curves to analyze entanglement in quantum states with conservation laws, providing explicit formulas for certain ensembles and exploring their properties.

Contribution

It extends the concept of Page curves to symmetry-resolved entanglement, deriving explicit formulas for Haar-random and fermionic Gaussian states with U(1) symmetry.

Findings

Symmetry-resolved Page curves can be derived analytically for Haar-random states.

For fermionic Gaussian states, an analytic thermodynamic limit result is obtained.

Numerical results confirm the theoretical predictions and finite-size effects are discussed.

Abstract

Given a statistical ensemble of quantum states, the corresponding Page curve quantifies the average entanglement entropy associated with each possible spatial bipartition of the system. In this work, we study a natural extension in the presence of a conservation law and introduce the symmetry-resolved Page curves, characterizing average bipartite symmetry-resolved entanglement entropies. We derive explicit analytic formulae for two important statistical ensembles with a $U(1)$-symmetry: Haar-random pure states and random fermionic Gaussian states. In the former case, the symmetry-resolved Page curves can be obtained in an elementary way from the knowledge of the standard one. This is not true for random fermionic Gaussian states. In this case, we derive an analytic result in the thermodynamic limit based on a combination of techniques from random-matrix and large-deviation theories. We test our predictions against numerical calculations and discuss the sub-leading finite-size corrections.