Holographic Duality Between Bulk Topological Order and Boundary Mixed-State Order

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 (SW-SSB) 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 SW-SSB 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.