Tensor Network Representations for Intrinsically Mixed-State Topological Orders

TL;DR

This paper introduces a method to construct tensor network representations for mixed-state topological phases, enabling analysis of complex quantum states with topological order affected by decoherence or disorder.

Contribution

The authors develop a general protocol leveraging anyon condensation in Choi states to represent mixed topological phases, including non-Abelian and chiral phases, which were previously difficult to model.

Findings

Constructed tensor network representations for various decohered topological codes.

Demonstrated applicability to non-Abelian and chiral topological phases.

Provided examples including $ ext{Z}_N$ toric code and $S_3$ quantum double.

Abstract

Tensor networks are an efficient platform to represent interesting quantum states of matter as well as to compute physical observables and information-theoretic quantities. We present a general protocol to construct fixed-point tensor network representations for intrinsically mixed-state topological phases, which exhibit nontrivial topological phenomena and do not have pure-state counterparts. The method exploits the power of anyon condensation in Choi states and is applicable to the cases where the target states arise from pure-state topological phases subject to strong decoherence/disorders in the Abelian sectors. Representative examples include $m^a e^b$ decoherence of $\mathbb{Z}_N$ toric code, decohered non-Abelian $S_3$ quantum double as well as pure $Z$/$X$ decoherence of arbitrary CSS codes. An example of chiral topological phases which cannot arise from local commuting projector models are also presented.