Climbing the Fusion Tree: Center Ordering and Decoding Hierarchy in Non-Abelian Mixed States

Abstract

We develop an operational approach to strong-to-weak spontaneous symmetry breaking (sw-SSB) of a non-Abelian symmetry group G. We formulate it as the decoding task of distinguishing two strongly symmetric states that differ by a pair of inserted irreps, after both are corrupted by strongly symmetric decoherence. For finite G, this task is dual to distinguishing logical states of the quantum double D(G) under pure-charge noise. We derive a hierarchy of statistical mechanics models describing optimal decoding from the fusion outcomes measured on successively larger nested regions. As for maximum-likelihood decoders of Abelian codes the measured irreps act as quenched disorder obeying a Nishimori-like condition. Strikingly, because local irreps do not determine how they fuse globally, the strong G× G symmetry is realized only nonlocally, and irreps trivially charged under Z(G) (the center of G) can hide inside noise-generated ones. As a result, the strong symmetry spontaneously breaks down to Z(G), times the weak symmetry. The only transition at nonzero error rate is then an ordering of the center, whose threshold cannot decrease as the hierarchy is ascended. This conclusion holds at any finite level of the hierarchy for non-nilpotent groups. We confirm these predictions with tensor-network calculations for S₃ and SU(2). At the first level, the S₃ model, whose center is trivial, shows no transition, and the fidelity correlations of all irreps provably decay to non-zero values. The SU(2) model exhibits a Z₂ transition consistent with the Nishimori-Ising universality class, where the fidelity correlations of spin-1/2 irreps become long-range. The threshold rises from the first to the second level of the hierarchy by 16%. Our results provide a systematic framework for studying non-Abelian sw-SSB and for constructing decoders for non Abelian topological codes.

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Cite

BibTeX RIS

BibTeX

@misc{sala2026climbing,
  title = {{Climbing the Fusion Tree: Center Ordering and Decoding Hierarchy in Non-Abelian Mixed States}},
  author = {Pablo Sala and Vlad Temkin and Cenke Xu and Daniel Podolsky and Ehud Altman},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06692},
  eprint = {2610.06692},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06692},
}

RIS

TY  - UNPB
TI  - Climbing the Fusion Tree: Center Ordering and Decoding Hierarchy in Non-Abelian Mixed States
AU  - Pablo Sala
AU  - Vlad Temkin
AU  - Cenke Xu
AU  - Daniel Podolsky
AU  - Ehud Altman
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06692
UR  - https://arxiv.org/abs/2610.06692
AB  - We develop an operational approach to strong-to-weak spontaneous symmetry breaking (sw-SSB) of a non-Abelian symmetry group G. We formulate it as the decoding task of distinguishing two strongly symmetric states that differ by a pair of inserted irreps, after both are corrupted by strongly symmetric decoherence. For finite G, this task is dual to distinguishing logical states of the quantum double D(G) under pure-charge noise. We derive a hierarchy of statistical mechanics models describing optimal decoding from the fusion outcomes measured on successively larger nested regions. As for maximum-likelihood decoders of Abelian codes the measured irreps act as quenched disorder obeying a Nishimori-like condition. Strikingly, because local irreps do not determine how they fuse globally, the strong G× G symmetry is realized only nonlocally, and irreps trivially charged under Z(G) (the center of G) can hide inside noise-generated ones. As a result, the strong symmetry spontaneously breaks down to Z(G), times the weak symmetry. The only transition at nonzero error rate is then an ordering of the center, whose threshold cannot decrease as the hierarchy is ascended. This conclusion holds at any finite level of the hierarchy for non-nilpotent groups. We confirm these predictions with tensor-network calculations for S_3 and SU(2). At the first level, the S_3 model, whose center is trivial, shows no transition, and the fidelity correlations of all irreps provably decay to non-zero values. The SU(2) model exhibits a Z_2 transition consistent with the Nishimori-Ising universality class, where the fidelity correlations of spin-1/2 irreps become long-range. The threshold rises from the first to the second level of the hierarchy by 16%. Our results provide a systematic framework for studying non-Abelian sw-SSB and for constructing decoders for non Abelian topological codes.
ER  -

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