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

TL;DR

This work establishes a holographic framework in which steady states of repeated quantum channels with strong symmetry map to boundary states of higher-dimensional topological orders via channel–state duality and isometric tensor networks (isoTNS). It shows that 1d SW-SSB in the steady state corresponds to boundary anyon condensation in a 2d bulk toric-code order, with the fidelity correlator and conditional mutual information interpreted as indicators of bulk–boundary anomalies. The authors develop continuously tunable isoTNS-dual channels, revealing a $U(1)$ SW-SSB critical point and finite-depth transitions, and extend the duality to a variety of generalized symmetries, including higher-form, subsystem, and fermionic cases. The framework provides a unifying viewpoint for mixed-state phases, offers explicit constructions of topological bulk waves that encode boundary SW-SSB, and suggests practical state-preparation protocols via isoTNS. Together, these results advance the understanding of how bulk topological order governs mixed-state boundary phenomena in nonequilibrium quantum dynamics.

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.

Figures (19)

• Figure 1: Holographic correspondence between an SW-SSB mixed state in $d$ dimensions and a topological order in $d+1$ dimensions. In a $\mathbb{Z}_2$ topological order, the two loop-like symmetries in the bulk can be pushed to the 1d boundary, where they manifest as a strong symmetry and a weak symmetry. The mutual anomaly between these symmetries signals an SW-SSB in the boundary reduced density matrix.
• Figure 2: (a) A repeated quantum channel acting on the system (wordline in red) in $d$ spatial dimensions is equivalent to sequentially entangling the system with fresh ancilla qubits in various time slices, which generates a pure-state wavefunction in $(d+1)$ spatial dimensions, as shown in (b). The system output under the repeated channel is the boundary reduced density matrix of this higher-dimensional wave function by tracing out the ancilla qubits.
• Figure 3: A repeated quantum channel can be viewed as a sequential unitary circuit where a 1d system initialized at $\ket{\psi_0}$ propagates upward and sequentially interacts row by row with ancilla qubits (shown in green and blue). Tracing out the green and blue qubits implements the X and ZZ noise channels, respectively. The reduced density matrix on the 1d top boundary (colored in red) represents the output of the repeated quantum channel. At $p_z = p_x = \frac{1}{2}$, the bulk has the fixed-point toric-code topological order, and the top boundary is the SW-SSB state, i.e. $\propto (1 + \prod_i X_i)$.
• Figure 4: (a) The inner circle and outer circle correspond to the top boundary and bottom boundary of a toric code defined on a cylinder. The reduced density matrix supported on the inner circle defines the 1d SW-SSB mixed state, which exhibits a non-zero conditional mutual information $I(A,C|B)$. (b) The conventional setup for defining the CMI (or equivalently, the topological entanglement entropy) of the toric code state. The nonzero CMI in both (a) and (b) configurations arises from the mutual anomaly of the 1-form symmetries in the 2D toric code topological order.
• Figure 5: The formation of the Pauli X string extending from the top boundary to the bottom boundary will modify the expectation value of $S_\gamma$ string. Only when $L_y\to \infty$, the probability of such an X string will vanish, leading to the robustness of the $S_\gamma$ string, and therefore, the emergence of SW-SSB on the top boundary.