Holographic duality between bulk topological order and boundary mixed-state order

TL;DR

This paper develops a holographic framework linking steady states of symmetric quantum channels to boundary states of higher-dimensional topological orders, revealing new insights into symmetry breaking and phase transitions.

Contribution

It introduces a novel holographic duality between quantum channel steady states and boundary states of topological orders, using tensor network states to analyze mixed-state phases.

Findings

Steady states of quantum channels are mapped to boundary reduced density matrices.

Strong-to-weak symmetry breaking is explained via anyon condensation.

Constructed tunable quantum channels with distinct phase transitions.

Abstract

We introduce a holographic framework for analyzing the steady states of repeated quantum channels with strong symmetries. Using channel-state duality, we show that the steady state of a $d$-dimensional quantum channel is holographically mapped to the boundary reduced density matrix of a $(d+1)$-dimensional wavefunction generated by a sequential unitary circuit. From this perspective, strong-to-weak spontaneous symmetry breaking (SWSSB) in the steady state arises from the anyon condensation on the boundary of a topological order in one higher dimension. The conditional mutual information (CMI) associated with SWSSB is then inherited from the bulk topological entanglement entropy. We make this duality explicit using isometric tensor network states (isoTNS) by identifying the channel's time evolution with the transfer matrix of a higher-dimensional isoTNS. Built on isoTNS, we further construct continuously tunable quantum channels that exhibit distinct mixed-state phases and transitions in the steady states.