Mixed-state Quantum Phases: Renormalization and Quantum Error Correction

TL;DR

This paper develops a real-space renormalization group framework for mixed quantum states, linking phase classification to quantum error correction, and applies it to analyze the phases of the toric code under various conditions.

Contribution

It introduces a novel RG scheme for mixed states based on local channels and connects phase classification with decodability in quantum error correction.

Findings

Finite temperature toric code flows to trivial phase under RG.

Local dephasing preserves the toric code phase.

Logical information cannot be destroyed without leaving the phase.

Abstract

Open system quantum dynamics can generate a variety of long-range entangled mixed states, yet it has been unclear in what sense they constitute phases of matter. To establish that two mixed states are in the same phase, as defined by their two-way connectivity via local quantum channels, we use the renormalization group (RG) and decoders of quantum error correcting codes. We introduce a real-space RG scheme for mixed states based on local channels which ideally preserve correlations with the complementary system, and we prove this is equivalent to the reversibility of the channel's action. As an application, we demonstrate an exact RG flow of finite temperature toric code in two dimensions to infinite temperature, thus proving it is in the trivial phase. In contrast, for toric code subject to local dephasing, we establish a mixed state toric code phase using local channels obtained by truncating an RG-type decoder and the minimum weight perfect matching decoder. We also discover a precise relation between mixed state phase and decodability, by proving that local noise acting on toric code cannot destroy logical information without bringing the state out of the toric code phase.