A note on invariants of mixed-state topological order in 2D
Yoshiko Ogata
TL;DR
This work addresses the classification of mixed-state topological order in two dimensions by identifying invariants that decrease under finite-depth quantum channels. It links the problem to a strict braided $C^*$-tensor category ${\mathcal D}_{\omega}$ arising from states with (approximate) Haag duality and defines the $S$-matrix $S^{\omega}$ and twist $\theta^{\omega}$ via the category's braiding. The main result shows that finite-depth channels induce matrices $X,Y,N$ with $S^{\omega_2}=X S^{\omega_1} Y$ and $\theta^{\omega_2}=N \theta^{\omega_1}$, along with dimension-relations $X d^{\omega_1}=d^{\omega_2}$ and $Y^{T} d^{\omega_1}=d^{\omega_2}$, yielding a preorder $S^{\omega_2} \prec_s S^{\omega_1}$ and $\theta^{\omega_2} \prec_t \theta^{\omega_1}$. This demonstrates that the $S$-matrix and twists serve as decreasing topological indices for mixed-state 2D systems, consistent with the notion that nontrivial abelian anyons cannot be created from product states by finite-depth mixing. The approach relies on a faithful unitary braided tensor functor between the relevant categories and a structural analysis of conjugates and traces.
Abstract
The classification of mixed-state topological order requires indices that behave monotonically under finite-depth quantum channels. In two dimensions, a braided $C^*$-tensor category, which corresponds to strong symmetry, arises from a state satisfying approximate Haag duality. In this note, we show that the $S$-matrix and topological twists of the braided $C^*$-tensor category are quantities that are monotone under finite-depth quantum channels.