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This research establishes a rigorous algebraic framework linking generalized symmetry breaking to information loss in topological quantum phases, specifically quantifying the entropic cost of anyon condensation via Watatani indices. By bridging operator algebras with quantum information theory, the study provides fundamental insights into the structural limits of coarse-graining observables in $(2+1)$-dimensional systems. These findings advance our theoretical understanding of non-invertible symmetries and their implications for fault-tolerant quantum computing architectures.
Information Loss in Generalized Symmetry Breaking
arXiv:2509.24625v4 Announce Type: replace Abstract: We present an algebraic and information-theoretic framework for the breaking of generalized, non-invertible symmetries in $(2+1)$ 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.
Abstract:We present an algebraic and information-theoretic framework for the breaking of generalized, non-invertible symmetries in $(2+1)$ 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.
v4: 31 pages, SciPost Submission. Revised version according to referee comments and reports. Results and conclusions unchanged. Improved presentation and additional discussion of the examples. References added
arXiv:2509.24625 [quant-ph]
(or arXiv:2509.24625v4 [quant-ph] for this version)
Full text under the publisher's open licence · 177 words
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