Resource-Theoretic Quantifiers of Weak and Strong Symmetry Breaking: Strong Entanglement Asymmetry and Beyond

TL;DR

This paper develops a resource theory framework to quantify and analyze strong symmetry breaking in quantum states, revealing how weak symmetry breaking can irreversibly convert into strong symmetry breaking, with applications to quantum field theory.

Contribution

It introduces a new resource theory for strong symmetry, identifying quantifiers like entanglement asymmetry and establishing a parallel with entanglement theory for U(1) symmetry.

Findings

Second-Rényi entanglement asymmetry is not a resource monotone.

The variance of conserved quantities characterizes strong symmetry breaking for U(1).

Framework connects weak and strong symmetry breaking in open quantum systems.

Abstract

Quantifying how much a quantum state breaks a symmetry is essential for characterizing phases, nonequilibrium dynamics, and open-system behavior. Quantum resource theory provides a rigorous operational framework to define and characterize such quantifiers of symmetry-breaking. As a starter, we exemplify the usefulness of resource theory by noting that second-R\'enyi entanglement asymmetry can increase under symmetric operations, and hence is not a resource monotone, and should not solely be used to capture Quantum Mpemba effect. More importantly, motivated by mixed-state physics where weak and strong symmetries are inequivalent, we formulate a new resource theory tailored to strong symmetry, identifying free states and strong-covariant operations. This framework systematically identifies quantifiers of strong symmetry breaking for a broad class of symmetry groups, including a strong entanglement asymmetry. A particularly transparent structure emerges for U(1) symmetry, where the resource theory for the strong symmetry breaking has a completely parallel structure to the entanglement theory: the variance of the conserved quantity fully characterizes the asymptotic manipulation of strong symmetry breaking. By connecting this result to the knowledge of the geometry of quantum state space, we obtain a quantitative framework to track how weak symmetry breaking is irreversibly converted into strong symmetry breaking in open quantum systems. We further propose extensions to generalized symmetries and illustrate the qualitative impact of strong symmetry breaking in analytically tractable QFT examples and applications.