Information Loss and Cost in Symmetry Breaking
Javier Molina-Vilaplana, Germán Sierra, H. C. Zhang
TL;DR
The paper develops an operator-algebraic framework to study symmetry breaking of generalized, non-invertible symmetries in two dimensions by embedding the problem in subfactor theory with condensable Frobenius algebras. It defines restriction (conditional expectation) and lifting (coarse-graining) maps that implement symmetry reduction and reconstruction, and introduces a relative-entropy order parameter $S_{\mathcal{A}}(\rho\|\tilde{\rho})$ that quantifies information loss, bounded by $\log\lambda$, where $\lambda=[\mathcal{A}:\mathcal{T}]$ equals the condensate quantum dimension $q$. The approach is demonstrated on concrete models including abelian $\mathbb{Z}_N$, the toric code, and Rep$(S_3)$, with dualities producing equivalent condensation patterns. By linking condensations to Morita equivalence and providing explicit Kraus representations, the work bridges operator algebras, tensor-category theory, and quantum information in the study of generalized symmetries and anyon condensation, offering a calculable, universal framework and suggesting directions for extensions to type II/III settings and lattice realizations.
Abstract
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.