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

Yuya Kusuki, Sridip Pal, Hiroyasu Tajima

TL;DR

This work develops a comprehensive resource-theoretic framework for strong symmetry breaking, identifying free states (strong symmetric and single-sector) and free operations (strong covariant) and introducing principled measures that distinguish weak from strong symmetry breaking. In the $U(1)$ case, the variance of the conserved quantity becomes an i.i.d.-complete quantifier that mirrors the entanglement entropy in entanglement theory, enabling exact interconversion rates between many copies of states. The authors extend these ideas to generalized symmetries and non-invertible algebras, provide physically transparent examples in CFT and spin chains, and connect the dynamics of symmetry breaking to the geometry of quantum state space, illuminating how weak symmetry can irreversibly convert into strong symmetry breaking in open systems. They also propose a dynamical cross-over, the Strong Mpemba effect, to diagnose symmetry dynamics beyond conventional equilibrium descriptions, with broad implications for nonequilibrium and 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-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.

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

TL;DR

This work develops a comprehensive resource-theoretic framework for strong symmetry breaking, identifying free states (strong symmetric and single-sector) and free operations (strong covariant) and introducing principled measures that distinguish weak from strong symmetry breaking. In the case, the variance of the conserved quantity becomes an i.i.d.-complete quantifier that mirrors the entanglement entropy in entanglement theory, enabling exact interconversion rates between many copies of states. The authors extend these ideas to generalized symmetries and non-invertible algebras, provide physically transparent examples in CFT and spin chains, and connect the dynamics of symmetry breaking to the geometry of quantum state space, illuminating how weak symmetry can irreversibly convert into strong symmetry breaking in open systems. They also propose a dynamical cross-over, the Strong Mpemba effect, to diagnose symmetry dynamics beyond conventional equilibrium descriptions, with broad implications for nonequilibrium and 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-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.
Paper Structure (45 sections, 24 theorems, 338 equations, 4 figures, 1 table)

This paper contains 45 sections, 24 theorems, 338 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction & Summary
  2. Preliminary: Basics of the Resource Theory of Asymmetry
  3. Resource theories
  4. Resource theory of asymmetry for weak symmetry
  5. Asymmetry monotones for weak symmetry
  6. Relative entropy of asymmetry
  7. Complete monotones for the i.i.d. convertibility
  8. Resource-theoretic Results for Weak Symmetry
  9. Rényi-2 proxies are not resource monotones
  10. Relative entropy of asymmetry and the thermodynamic limit
  11. Resource Theory of Asymmetry for Strong Symmetry
  12. Free states and free operations
  13. Physical realization and Kraus representation of strong covariant operations
  14. Relation between strong covariant and weak covariant
  15. Resource measures
  16. ...and 30 more sections

Key Result

Theorem 1

Let $G$ be a group, and let $U:G\rightarrow U({\cal H})$ and $U':G\rightarrow U'({\cal H}')$ be unitary representations acting on ${\cal H}$ and ${\cal H}'$. Let a CPTP map $\Lambda:{\cal B}({\cal H})\rightarrow {\cal B}({\cal H}')$ be a $(U,U')$-covariant operation. Then, there exist two auxiliary Conversely, when a CPTP map $\Lambda:{\cal B}({\cal H})\rightarrow {\cal B}({\cal H}')$ can be real

Figures (4)

  • Figure 1: Classifications of states. In the figure, asymmetric states are indicated by the hatched region, whereas symmetric states are shown as the three shaded regions. The set of symmetric states exhibits a three-level nested hierarchical structure, i.e. (strong symmetric)$\subset$(single-sector)$\subset$(weak symmetric). The important features of this structure is the following four: (a) There exist cases where (strong symmetric)$\subsetneq$(single-sector) and (single-sector)$\subsetneq$(weak symmetric) hold. (b) When $G=U(1)$, (strong symmetric)$=$(single-sector) holds. (c) When $G$ is some non-Abelian group, sometimes there are no strong symmetric states. (d) For any $G$, there is at least a single-sector state.
  • Figure 2: Schematic diagram of physical realization of operations that are strong covariant with respect to particle number and weak covariant with respect to energy. The system exchanges energy with an external heat bath, while no particles are exchanged.
  • Figure 3: Dashed blue line denotes the evolution of strong asymmetry for $\rho_1$ and the thick black line denotes the evolution of strong asymmetry for $\rho_2$. There is a cross-over under evolution by \ref{['master']} (here $k=1$). See eqs. \ref{['eq:r']}, \ref{['eq:Asss']} and \ref{['eq:defF']}.
  • Figure 4: Dashed blue line denotes the evolution of weak asymmetry for $\rho_1$ and the thick black line denotes the evolution of weak asymmetry for $\rho_2$. The evolution happens according to \ref{['master']} (here $k=1$). Both asymptotes to $0$ without having any cross-over. See eqs. \ref{['eq:r']}, \ref{['eq:Asss']} and \ref{['eq:defF']}.

Theorems & Definitions (45)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • Definition 1: Entanglement asymmetry of strong symmetry breaking
  • Theorem 6
  • Definition 2: Averaged logarithmic characteristic function
  • Theorem 7
  • ...and 35 more