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arxiv: 2601.20924 · v2 · submitted 2026-01-28 · ✦ hep-th · cond-mat.stat-mech· quant-ph

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

Yuya Kusuki , Sridip Pal , Hiroyasu Tajima This is my paper

Pith reviewed 2026-05-16 10:17 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mechquant-ph
keywords symmetry breakingresource theoryentanglement asymmetryU(1) symmetryopen quantum systemsmixed statesstrong symmetryquantum field theory
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0 comments X

The pith

For U(1) symmetry the variance of the conserved quantity fully characterizes asymptotic manipulation of strong symmetry breaking.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a resource theory to quantify strong symmetry breaking in quantum states, which is inequivalent to weak symmetry breaking in mixed-state settings. It defines free states and strong-covariant operations that respect strong symmetry, yielding new quantifiers including a strong entanglement asymmetry. For U(1) symmetries this framework mirrors the structure of entanglement theory, with the variance of the conserved charge determining what asymptotic manipulations are possible. The resulting picture supplies a quantitative way to track irreversible conversion from weak to strong symmetry breaking in open quantum systems.

Core claim

Motivated by the inequivalence of weak and strong symmetries in mixed-state physics, we formulate a resource theory for strong symmetry breaking by identifying free states and strong-covariant operations. For U(1) symmetry the resource theory has a completely parallel structure to the entanglement theory, where the variance of the conserved quantity fully characterizes the asymptotic manipulation of strong symmetry breaking. By connecting this result to 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.

What carries the argument

The resource theory of strong symmetry breaking, defined via strong-covariant operations and free states that are invariant under strong symmetry; for U(1) this reduces asymptotic rates exactly to the variance of the conserved quantity.

If this is right

  • Second-Rényi entanglement asymmetry can increase under symmetric operations and therefore is not a resource monotone.
  • Strong symmetry breaking admits quantifiers for a broad class of symmetry groups, with extensions proposed for generalized symmetries.
  • The framework supplies a quantitative way to track irreversible conversion of weak symmetry breaking into strong symmetry breaking in open quantum systems.
  • Qualitative effects of strong symmetry breaking appear in analytically tractable quantum field theory examples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same variance characterization may simplify numerical estimates of symmetry-breaking rates in complex many-body states.
  • The conversion tracking could be tested in open-system experiments that prepare states with controlled weak versus strong symmetry.
  • Links to state-space geometry suggest possible bounds on how fast strong symmetry can emerge from weak symmetry under realistic noise.

Load-bearing premise

The chosen strong-covariant operations and free states correctly capture the physical distinction between weak and strong symmetries.

What would settle it

An explicit calculation for a U(1)-symmetric mixed state showing that the asymptotic rate for manipulating strong symmetry breaking is not determined solely by the variance of the conserved charge.

Figures

Figures reproduced from arXiv: 2601.20924 by Hiroyasu Tajima, Sridip Pal, Yuya Kusuki.

Figure 1
Figure 1. Figure 1: FIG. 1. Classifications of states. In the figure, asymmetric [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic diagram of physical realization of opera [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Dashed blue line denotes the evolution of strong [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Dashed blue line denotes the evolution of weak asym [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops a resource theory for strong symmetry breaking in quantum systems, motivated by the distinction between weak and strong symmetries in mixed-state physics. It shows that the second Rényi entanglement asymmetry is not a monotone under symmetric operations, defines free states and strong-covariant operations for a broad class of symmetry groups, and establishes a strong entanglement asymmetry quantifier. For U(1) symmetry, the framework parallels entanglement theory, with the variance of the conserved charge serving as the complete asymptotic monotone for state manipulation; this is connected to the geometry of quantum state space to describe irreversible conversion from weak to strong symmetry breaking in open systems, with extensions to generalized symmetries and QFT examples.

Significance. If the central constructions hold, the work supplies an operational, resource-theoretic foundation for distinguishing weak and strong symmetry breaking, with direct implications for nonequilibrium dynamics, open quantum systems, and quantum phases. The U(1) result, by reducing asymptotic convertibility to charge variance and linking to known state-space geometry, offers a quantitative handle on irreversible processes that could be tested in concrete models; the parallel to entanglement theory is a notable strength, as it imports existing asymptotic results without introducing new free parameters.

major comments (2)
  1. [Resource theory formulation] The section introducing strong-covariant operations and free states: the operational distinction between weak and strong symmetries is load-bearing for all subsequent claims, yet the manuscript provides only a high-level motivation from mixed-state physics without a concrete counter-example showing that standard covariant operations fail to capture the strong case.
  2. [U(1) case] U(1) symmetry section on asymptotic manipulation: the claim that variance of the conserved quantity fully characterizes convertibility rates is central, but the derivation appears to rely on an unstated extension of the majorization or convex-roof construction from entanglement theory; an explicit statement of the monotone set and the proof that no additional independent monotones exist is required.
minor comments (2)
  1. [Introduction] The abstract states that second-Rényi entanglement asymmetry can increase under symmetric operations; a brief numerical example or explicit channel in the main text would make this observation more accessible.
  2. [Definitions] Notation for the strong entanglement asymmetry functional should be introduced with a clear comparison table to the ordinary entanglement asymmetry to avoid reader confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation and constructive feedback on our manuscript. The comments help clarify key aspects of the resource theory construction. We address each major comment below and outline the revisions.

read point-by-point responses

  1. Referee: [Resource theory formulation] The section introducing strong-covariant operations and free states: the operational distinction between weak and strong symmetries is load-bearing for all subsequent claims, yet the manuscript provides only a high-level motivation from mixed-state physics without a concrete counter-example showing that standard covariant operations fail to capture the strong case.

    Authors: We agree that an explicit counter-example strengthens the operational motivation. While the distinction is grounded in established mixed-state physics literature, the revised manuscript will include a concrete example in the section on strong-covariant operations: consider a U(1)-symmetric mixed state obtained by depolarizing a coherent state in the charge basis; standard covariant operations preserve the weak symmetry (commuting with the charge operator) but cannot enforce the strong symmetry constraint on the support of the state, whereas strong-covariant operations do. This example will be added with explicit calculations to illustrate the failure of standard operations. revision: yes

  2. Referee: [U(1) case] U(1) symmetry section on asymptotic manipulation: the claim that variance of the conserved quantity fully characterizes convertibility rates is central, but the derivation appears to rely on an unstated extension of the majorization or convex-roof construction from entanglement theory; an explicit statement of the monotone set and the proof that no additional independent monotones exist is required.

    Authors: We acknowledge that the asymptotic characterization requires a more explicit derivation. The variance serves as the complete monotone because, for U(1) charges, the majorization relation on the charge distribution reduces to the variance under the convex-roof extension. In the revision, we will add an explicit statement of the monotone set (variance plus subadditive functions that are constant on free states) and a self-contained proof that no additional independent monotones exist for asymptotic rates, adapting the standard majorization arguments from entanglement theory to the charge sector while citing the relevant geometric results on state space. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines a resource theory for strong symmetry breaking using standard axioms of quantum resource theories, identifying free states and strong-covariant operations directly from the physical distinction between weak and strong symmetries. For U(1), the variance of the conserved charge emerges as the asymptotic monotone from the structure of the theory itself, paralleling entanglement theory without any reduction to fitted inputs, self-citations, or renamed known results. The connection to quantum state space geometry is invoked as external knowledge, not derived internally. No load-bearing self-citation chains or ansatzes smuggled via prior work are present in the provided framework.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the standard axioms of quantum resource theory, the physical inequivalence of weak and strong symmetries in mixed states, and the definition of strong-covariant operations as free operations.

axioms (2)
  • standard math Standard axioms of quantum resource theory for defining free states and operations
    The paper explicitly builds a new resource theory by identifying free states and strong-covariant operations.
  • domain assumption Weak and strong symmetries are inequivalent in mixed-state physics
    Motivation stated directly in the abstract for formulating the strong symmetry resource theory.
invented entities (1)
  • strong entanglement asymmetry no independent evidence
    purpose: Quantifier of strong symmetry breaking
    New monotone introduced within the strong symmetry resource theory framework.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Charge Scrambling in Strong-to-Weak Spontaneous Symmetry Breaking

    cond-mat.stat-mech 2026-05 unverdicted novelty 7.0

    Long-range Rényi-1 SWSSB order implies extensive block-charge variance for continuous symmetries with rapid asymptotic approach, with conditional counterexamples and a new twist overlap correlator separating symmetry ...

  2. Enhancing entanglement asymmetry in fragmented quantum systems

    cond-mat.stat-mech 2026-03 unverdicted novelty 6.0

    Entanglement asymmetry for inhomogeneous U(1) charges in fragmented systems scales extensively, is bounded by a universal fraction of its maximum, and distinguishes classical from quantum fragmentation.

Reference graph

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