A note on invariants of mixed-state topological order in 2D

TL;DR

This paper explores how certain invariants like the S-matrix and topological twists in 2D mixed-state topological order behave monotonically under finite-depth quantum channels, aiding classification.

Contribution

It demonstrates that the S-matrix and topological twists are monotone invariants under finite-depth quantum channels in 2D mixed-state topological order.

Findings

S-matrix is monotone under finite-depth quantum channels

Topological twists are monotone under finite-depth quantum channels

Provides tools for classifying mixed-state topological phases

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.