Information Loss in Generalized Symmetry Breaking

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 $C^{\ast }$-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.

arXiv commentv4: 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