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arxiv: 2509.24179 · v2 · submitted 2025-09-29 · 🪐 quant-ph

Strong-to-weak spontaneous symmetry breaking of higher-form non-invertible symmetries in Kitaev's quantum double model

Zijian Song , Jian-Hao Zhang This is my paper

Pith reviewed 2026-05-18 13:33 UTC · model grok-4.3

classification 🪐 quant-ph
keywords strong-to-weak spontaneous symmetry breakingnon-invertible higher-form symmetriesKitaev quantum double modeldecoherencemixed statesinformation convex settopological orderground-state degeneracy
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The pith

Decoherence in non-Abelian Kitaev quantum double models converts strong non-invertible higher-form symmetries to weak ones while the information convex set of mixed states has dimension equal to the pure-state ground-state degeneracy.

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

The paper investigates spontaneous symmetry breaking of higher-form symmetries in the setting of mixed states obtained by decohering non-Abelian Kitaev quantum double models. It establishes that these symmetries undergo strong-to-weak spontaneous symmetry breaking, after which the decohered states remain locally indistinguishable and form an information convex set. The central result is that the dimension of this convex set exactly equals the ground-state degeneracy of the corresponding pure state. This equivalence shows how quantum information stored in the ground-state subspace is converted into classical information stored in the set of decohered density matrices. A reader would care because the construction links the classification of topological order in open systems to concrete properties of convex sets of mixed states.

Core claim

Under decoherence the non-Abelian Kitaev quantum double models exhibit strong-to-weak spontaneous symmetry breaking of non-invertible higher-form symmetries; the resulting mixed states form a locally indistinguishable set that is also an information convex set, and the dimension of this convex set equals the ground-state degeneracy of the pure state, so that quantum information in the ground-state subspace is degraded into classical information captured by the convex set of decohered density matrices.

What carries the argument

the information convex set of decohered density matrices, whose dimension is shown to equal the ground-state degeneracy of the pure state

Load-bearing premise

The decoherence process preserves local indistinguishability while converting non-invertible higher-form symmetries from strong to weak, so that the information convex set stays well-defined and its dimension matches the pure-state degeneracy.

What would settle it

An explicit calculation of the dimension of the information convex set for a concrete non-Abelian Kitaev quantum double model after a specific decoherence channel that yields a value different from the pure-state ground-state degeneracy would falsify the central claim.

Figures

Figures reproduced from arXiv: 2509.24179 by Jian-Hao Zhang, Zijian Song.

Figure 1
Figure 1. Figure 1: In particular, a ribbon ξ consists of a sequence of direct and dual triangles along the path. For ribbon op￾erators of the form F e,g ξ , the action on every dual triangle τ ∗ ∈ ξ is trivial, allowing us to reformulate the ribbon operators in terms of generalized Z operators, namely F e,g ξ = 1 |G| X Γ∈Rep(G) dΓ · tr  Γ(¯g) Y τ∈ξ Z ± Γ,τ   , (7) where the product is taken over all qudits on the direct … view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. ’t Hooft anomaly/SSB of 1-form ribbon symmetries. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. A tripartition [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

Topological orders can be understood as spontaneous symmetry breaking of higher-form symmetries. In the non-Abelian case, the broken higher-form symmetries are notably non-invertible. In this work, we extend this framework to mixed states, where symmetries can be either strong or weak. In particular, we investigate the strong-to-weak spontaneous symmetry breaking (SWSSB) of non-invertible higher-form symmetries in non-Abelian Kitaev's quantum double models under decoherence. We further show that the resulting decohered quantum double mixed states form a locally indistinguishable set, which also constitutes an information convex set. Importantly, we emphasize that the dimension of this convex set equals the ground-state degeneracy of the corresponding pure state, highlighting that the quantum information encoded in the ground-state subspace is degraded into classical information captured by the convex set of decohered density matrices.

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 extends the spontaneous symmetry breaking framework for topological orders to mixed states by studying strong-to-weak spontaneous symmetry breaking (SWSSB) of non-invertible higher-form symmetries in non-Abelian Kitaev quantum double models under decoherence. It constructs decohered mixed states that form a locally indistinguishable set, shows this set is an information convex set, and claims that the dimension of the convex set equals the ground-state degeneracy of the corresponding pure state, thereby interpreting the process as degradation of quantum information into classical information captured by the convex set.

Significance. If the central claims hold, the work provides a symmetry-based characterization of topological order in decohered non-Abelian systems and a direct link between pure-state degeneracy and the structure of the information convex set. This could inform studies of mixed-state phases and quantum information degradation under noise, with the concrete Kitaev double example serving as a useful test case.

major comments (2)
  1. [§4] §4 (decoherence channel construction): The map converting strong non-invertible higher-form symmetries to weak ones is defined via its action on symmetry operators and fusion data, but there is no explicit verification that the channel commutes with the relevant fusion rules or that the resulting mixed states remain locally indistinguishable on contractible regions. This preservation is required for the information convex set to be well-defined and for its dimension to equal the pure-state GSD.
  2. [§5.2] §5.2 (dimension equality argument): The claim that dim(information convex set) equals the ground-state degeneracy relies on the decohered states satisfying the local indistinguishability criterion used to define the convex set; without a direct check that this criterion survives the non-invertible symmetry action under the chosen noise channel, the equality does not follow from the preceding constructions.
minor comments (2)
  1. [§2] The notation for the fusion category data and the precise definition of 'weak' versus 'strong' symmetry in the mixed-state setting could be introduced with a short example from the Abelian case before the non-Abelian discussion.
  2. [§5.1] A few sentences clarifying how the information convex set is constructed from the decohered density matrices (e.g., via the explicit convex combination) would improve readability in §5.1.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address the two major comments point by point below. Where the referee correctly identifies the need for additional explicit verification, we have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [§4] §4 (decoherence channel construction): The map converting strong non-invertible higher-form symmetries to weak ones is defined via its action on symmetry operators and fusion data, but there is no explicit verification that the channel commutes with the relevant fusion rules or that the resulting mixed states remain locally indistinguishable on contractible regions. This preservation is required for the information convex set to be well-defined and for its dimension to equal the pure-state GSD.

    Authors: We thank the referee for this observation. The decoherence channel is defined to act on the symmetry operators and fusion data in a manner that preserves the algebraic structure of the non-invertible higher-form symmetries by construction. To make this explicit, we will add a direct verification in the revised §4 showing that the channel commutes with the fusion rules, using the representation theory of the quantum double. We will also include an explicit check that the resulting mixed states remain locally indistinguishable on contractible regions, by demonstrating that the reduced density matrices on such regions coincide for all states in the set. These additions ensure the information convex set is rigorously well-defined. revision: yes

  2. Referee: [§5.2] §5.2 (dimension equality argument): The claim that dim(information convex set) equals the ground-state degeneracy relies on the decohered states satisfying the local indistinguishability criterion used to define the convex set; without a direct check that this criterion survives the non-invertible symmetry action under the chosen noise channel, the equality does not follow from the preceding constructions.

    Authors: We agree that strengthening the argument with an explicit check is beneficial. In the revised §5.2 we will add a direct demonstration that the local indistinguishability criterion is preserved under the action of the non-invertible symmetries after the noise channel is applied. This follows from the channel commuting with the symmetries (as verified in the updated §4) and from the topological nature of the decoherence. With this check in place, the equality between the dimension of the information convex set and the pure-state ground-state degeneracy follows directly, confirming the interpretation of quantum information degradation into classical information. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on external definitions of information convex sets and symmetry breaking without self-referential reduction

full rationale

The abstract presents the dimension of the information convex set equaling pure-state GSD as a result of analyzing decohered states under strong-to-weak symmetry breaking in Kitaev quantum doubles. No equations, fitted parameters, or self-citations are visible that would make this equality hold by construction or via load-bearing self-reference. The local indistinguishability and convex-set properties are invoked as preserved under the decoherence map, but this is framed as a shown property rather than a definitional tautology. The derivation chain therefore remains self-contained against the stated assumptions and does not reduce to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities can be extracted. The framework implicitly assumes standard properties of Kitaev quantum double models and decoherence channels.

pith-pipeline@v0.9.0 · 5681 in / 1101 out tokens · 37169 ms · 2026-05-18T13:33:41.115788+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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  1. Universal Dynamical Scaling of Strong-to-Weak Spontaneous Symmetry Breaking in Open Quantum Systems

    cond-mat.mes-hall 2026-03 unverdicted novelty 7.0

    Symmetry class alone sets SWSSB correlation length growth to exponential (Z2, tc ~ ln L) or algebraic (U(1), tc ~ L^alpha with alpha filling-dependent) in open quantum systems, independent of spectral gap.

Reference graph

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