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Anyon condensation is encoded as a conditional expectation between operator algebras, and the information it erases, measured by relative entropy, is claimed to be bounded by the log of the condensate's quantum dimension.

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Anyon condensation is encoded as a conditional expectation between operator algebras, and the information it erases, measured by relative entropy, is claimed to be bounded by the log of the condensate's quantum dimension.

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arxiv 2509.24625 v3 pith:SPZU7YJB submitted 2025-09-29 quant-ph hep-thmath-phmath.MP

Information Loss in Generalized Symmetry Breaking

Javier Molina-Vilaplana , Germ\'an Sierra , H.C. Zhang This is my paper
classification quant-ph hep-thmath-phmath.MP
keywords generalizedinformationquantumalgebrasbreakingcondensedconditionalentropy
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

Topological phases of matter in two dimensions host anyons, quasiparticles with exotic exchange statistics. Sometimes a subset of anyons can condense: a new vacuum forms and the set of distinct particle types shrinks. This is a symmetry-breaking process for generalized, non-invertible symmetries, which cannot be described by ordinary group theory. The paper's proposal is to describe this process the way a quantum communication engineer would: as a quantum channel that sends the algebra of observables of the original phase into the smaller algebra of the condensed phase. The channel is built from branching coefficients, the data that say how each original particle type decomposes into condensed-phase types. A second map, the lifting map, tries to undo the damage by embedding the condensed state back into the original algebra.

The information lost in the round trip is measured by the relative entropy between the original state and its lifted image. Because of superselection rules, the paper restricts attention to states diagonal in particle type, so the whole computation reduces to classical probability vectors and the "quantum" relative entropy is just the Kullback-Leibler divergence. The central quantitative claim, equation (3.33), is that this information loss never exceeds the logarithm of the Jones index of the inclusion, which equals the quantum dimension of the condensate. The paper does not prove this bound; it states it and verifies it in examples: the group Z_N, the toric code, and the representation category Rep(S_3).

The worked examples are consistent with the bound, but one displayed formula, equation (4.15) for the Rep(S_3) condensation 1+Y, contradicts the example values printed two lines below it. The examples themselves are correct; the formula is not.

Core claim

The universal bound S_A(rho|rho_tilde) <= log lambda, Eqs. (3.33): the relative entropy between an anyonic state and its image under restriction followed by lifting is bounded by the logarithm of the Jones index lambda = [A:T], which equals the quantum dimension q of the condensate. The abstract states the paper 'establishes a universal bound governed by the Jones index, which is equal to the quantum dimension of the condensate.' If correct, every condensation pattern's information loss is capped by a single algebraic invariant of the inclusion.

Load-bearing premise

That the Pimsner-Popa style index bound applies to the abelian inclusion T subset A with the specific lifting map alpha, so that S(rho|alpha o epsilon(rho)) <= log lambda (Eq. 3.33) holds for 'well-defined probability distributions.' This is asserted in one sentence with no derivation; the cited Pimsner-Popa result (Eq. 3.4) bounds the Connes-Stormer entropy H(M|N), a different functional, and that theorem concerns inclusions of factors, whereas every example uses abelian, non-factor algebras. The paper also implicitly assumes the orthogonality of branching coefficients sum_a n^s_a n^t_a = delta^{st} q d_t, used in the idempotence proof in Appendix A but never stated.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The framework introduces no new physical entities and fits no free parameters to data. Its load-bearing inputs are: the branching coefficients of anyon condensation imported from [46-48] (with [48] co-authored by Sierra), the superselection-based restriction to abelian algebras, an unstated orthogonality relation needed for idempotence, and the Pimsner-Popa inequality whose direct applicability to the central bound is asserted, not demonstrated. The initial probability distributions chosen in the examples are inputs used only for illustration.

axioms (5)
  • standard math Standard von Neumann algebra and relative entropy background of Section 2: normal states as density matrices, relative entropy via (2.8).
    Background formalism, textbook material, not load-bearing beyond setting notation.
  • domain assumption Branching coefficients n^t_a with dimension constraints d_a = sum_t n^t_a d_t and d_t = (1/q) sum_a n^t_a d_a, Eqs. (3.8), imported from Bais-Slingerland and Neupert et al. [46-48].
    The entire channel construction rests on these condensation data from prior literature; [48] is co-authored by one of the present authors, though (3.8) is independently checkable and holds in all examples.
  • domain assumption Anyonic superselection rules justify restriction to an abelian algebra A with rho = sum_a p_a Pi_a (Section 3.1).
    This reduces the framework to classical probability vectors and drops the internal multiplicity spaces of non-abelian anyons (e.g., the 2-dimensional Y sector of Rep(S_3)); the advertised finite-dimensional C*-algebra generality is not implemented.
  • domain assumption Orthogonality of branching coefficients sum_a n^s_a n^t_a = delta^{st} q d_t, used in the idempotence proof in Appendix A.
    Needed for epsilon_A^2 = epsilon_A (Eqs. A.1-A.3); stated nowhere in the paper, follows from consistency of (3.8) but is not derived.
  • standard math Pimsner-Popa inequality H(M|N) <= log[M:N], Eq. (3.4), ref. [43].
    Cited as the conceptual basis for the central bound (3.33), but no bridge between the Connes-Stormer entropy H(M|N) and the two-state relative entropy S(rho|rho_tilde) is given.

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Pith. "Pith review of Information Loss in Generalized Symmetry Breaking." pith.science (2026). https://pith.science/paper/SPZU7YJB

@misc{pith2026250924625,
  author       = {Pith},
  title        = {Pith review of: Information Loss in Generalized Symmetry Breaking},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SPZU7YJB}},
  note         = {Machine review of arXiv:2509.24625}
}
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abstract

We present an algebraic and information-theoretic framework for the breaking of generalized, non-invertible symmetries in two spatial dimensions. Such patterns are modeled as inclusions of finite-dimensional $C^*$-algebras equipped with conditional expectations, built upon a precise dictionary with anyon condensation in topological phases of matter. The conditional expectations are quantum channels that coarse-grain observables of the parent phase onto the symmetry-reduced condensed phase; their index -- a Watatani index equal to the quantum dimension of the condensate -- bounds, through its logarithm, the relative entropy between a state and its condensed image. This relative entropy serves as an entropic order parameter quantifying the information lost in the symmetry-reduction transition. We illustrate the framework with explicit examples: the toric code, abelian groups $Z_N$, and the representation category Rep$(S_3)$. Our results strengthen the connections between operator algebras and quantum information in the study of generalized symmetries.

Figures

Figures reproduced from arXiv: 2509.24625 by Germ\'an Sierra, H.C. Zhang, Javier Molina-Vilaplana.

Figure 1
Figure 1. Figure 1: Principal graph for the symmetry breaking pattern induced by the algebra [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Principal graph for the symmetry breaking pattern induced by the algebra [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Principal graph for the symmetry breaking pattern induced by the algebra [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Principal graph for the symmetry breaking pattern induced by the algebra [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Principal graph for the symmetry breaking pattern induced by the algebra [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Principal graph for the symmetry breaking pattern induced by the algebra [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗

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

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    Using SymTFT, the entropic order parameter for a symmetry-breaking vacuum labelled by a equals log(dim C / d_a^2), making the distinguishability of non-invertible vacua manifest.

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.