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

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-Rényi 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.

💡 Research Summary

This paper addresses the fundamental problem of quantifying how much a quantum state breaks a given symmetry, a question that is central to the study of phases of matter, nonequilibrium dynamics, and open‑system behavior. While many physical works rely on order parameters, correlation functions, or heuristic “entanglement asymmetry” measures, the authors argue that only quantities derived from a rigorous resource‑theoretic framework can serve as reliable monotones—functions that never increase under the set of free (symmetry‑preserving) operations.

The first technical contribution is a careful analysis of the second‑Rényi entanglement asymmetry. Although this quantity is computationally convenient, the authors demonstrate explicitly that it can increase under symmetric CPTP maps, violating the monotonicity axiom required of any resource monotone. Consequently, conclusions drawn solely from the second‑Rényi proxy—particularly in studies of the quantum Mpemba effect—may be misleading.

Motivated by recent developments in open‑quantum‑system theory, the authors distinguish two notions of symmetry for mixed states: weak symmetry, defined by invariance under conjugation (U_g\rho U_g^\dagger=\rho), and strong symmetry, defined by a stricter condition (U_g\rho=e^{i\theta_g}\rho). Every strong‑symmetric state is automatically weak‑symmetric, but the converse fails in general, especially for non‑abelian groups where strong‑symmetric states may not exist at all. Existing asymmetry measures (including the standard entanglement asymmetry) are tailored to weak symmetry and cannot detect the breaking of strong symmetry.

To fill this gap, the authors construct a resource theory of strong symmetry. The free states (\mathcal{F}{\text{strong}}) are precisely those that satisfy the strong symmetry condition. Free operations are defined as strong‑covariant channels, i.e. CPTP maps whose Kraus operators all transform with the same group‑dependent phase, guaranteeing that free states are mapped to free states. For non‑abelian groups a middle‑tier class of “single‑sector states” is introduced, which relaxes the strong condition while still being stricter than weak symmetry. The paper proves that the set ((\mathcal{F}{\text{strong}},\text{strong‑covariant})) satisfies all the axioms of a consistent resource theory.

Within this framework the authors propose several strong‑symmetry monotones. The most transparent example is for the abelian group (U(1)). Here the variance of the conserved charge (Q), \