Information Loss in Generalized Symmetry Breaking

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.