Information Loss and Cost in Symmetry Breaking

We develop an algebraic and information-theoretic framework to characterize symmetry breaking of generalized, non-invertible symmetries in two spatial dimensions. The reduction of symmetry is modeled within subfactor theory, where condensable Frobenius algebras play the role of subgroups in the categorical setting. This perspective naturally connects to the description of anyon condensation in topological phases of matter. Central to our approach are coarse-graining maps, or conditional expectations, which act as quantum channels projecting observables from a phase with higher symmetry onto one where the symmetry is partially or completely broken by condensation. By employing relative entropy as an entropic order parameter, we quantify the information loss induced by condensation and establish a universal bound governed by the Jones index, which is equal to the quantum dimension of the condensate. We illustrate the framework through explicit examples, including the toric code, abelian groups $\mathbb{Z}_N$, and the representation category Rep($S_3$), and show how dualities give rise to equivalence classes of condensation patterns. Our results forge new connections between operator algebras, tensor category theory, and quantum information in the study of generalized symmetries.