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 $G\times G$ 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.

Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el)

arXiv:2610.06692 [quant-ph]

(or arXiv:2610.06692v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.06692 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.