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arXiv:2606.22507 · Physical review. B./Physical review. B

Hierarchy of mixed symmetry protected topological states in extended cluster states under subsystem decoherence

Yoshihito Kuno, Takahiro Orito

quant-phcond-mat.stat-mechcond-mat.str-elSubmitted Jun 21, 2026• Announced Jun 23, 2026• 0 citations
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Abstract

We study the effect of subsystem decoherence to an extended cluster state which is a symmetry protected topological (SPT) phase. The model includes many subsystem $Z_2$ symmetries. We report that subsystem decoherence induces local charge fluctuations, leading to a mixed SPT state in the unaffected subsystems. If we start from the extended cluster state, hierarchical mixed-state SPT phases emerge in response to step-by-step subsystem decoherences. These mixed-state SPT phases keep strong symmetries the symmetry of which is protecting symmetries for the initial cluster SPT. Moreover, these SPTs can be characterized by Rényi-2 string orders. Then, as the subsystems are progressively decohered, the hierarchy of mixed-state SPT phases terminates in a $Z_2$ strong-to-weak spontaneous symmetry breaking (SWSSB) state on the final remaining subsystem, where a long-range entangled state appears, namely a glassy Greenberger-Horne-Zeilinger (GHZ) state. Our work demonstrates that decoherence is not merely a destructive process, but can induce and organize series of nontrivial mixed-states. This reveals a systematic route from mixed-state SPT order to SWSSB with the glassy GHZ-type long-range entanglement.

Comments
16 page, 5 figures

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We explore the effect of decoherence on many-body mixed states from a purification perspective. Here, quantum channels map to unitary transformations acting on a purified state defined within an extended Hilbert space. First, we exploit this approach to analyze the phenomenon of spontaneous strong-to-weak symmetry breaking (sw-SSB) triggered by strongly symmetric local quantum channels. We find that sw-SSB for mixed states relates to proximity to a symmetry-protected topological (SPT) order in the purified state. Remarkably, the measurement-induced long-range order in the purified SPT state, as characterized by the R\'enyi-2 correlator, mirrors the ensuing long-range order in the mixed state due to sw-SSB. Moreover, we establish a 1-to-1 correspondence between various sw-SSB order parameters for the mixed state, and strange correlators in the purification, which signify the SPT order. This purification perspective is further extended to explore intrinsic mixed-state topological order and decoherent symmetry-protected topological phases.
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We investigate mixed states exhibiting nontrivial topological features, focusing on symmetry-protected topological (SPT) phases under various types of decoherence. Our findings demonstrate that these systems can retain topological information from the SPT ground state despite decoherence. In the ''doubled Hilbert space,'' we define symmetry-protected topological ensembles (SPT ensembles) and examine boundary anomalies in this space. We generalize the concept of the strange correlator, initially used to diagnose SPT ground states, to identify anomalies in mixed-state density matrices. Through exact calculations of stabilizer Hamiltonians and field theory evaluations, we show that nontrivial features of SPT states persist in two types of strange correlators: type-I and type-II. The type-I strange correlator reveals SPT information that can be efficiently detected and used experimentally, such as in preparing long-range entangled states. The type-II strange correlator encodes the full topological response of the decohered mixed state, reflecting the SPT state's pre-decoherence presence. Our work offers a unified framework for understanding decohered SPT phases from an information-theoretic perspective.
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Mixed-state phases have recently attracted significant attention as a generalization beyond their pure-state counterparts. Prominent examples include mixed-state symmetry-protected topological (mSPT) phases and the strong-to-weak symmetry breaking (SWSSB) phases. It has been shown recently that mSPT phases admit a holographic dual description in terms of higher-order subsystem SPT phases. In this work, we investigate the mixed-state phases obtained by tracing out the bulk degrees of freedom of higher-order subsystem SPT phases protected by non-invertible symmetries. We find that the resulting mixed states exhibit the coexistence of the symmetry-protected topological order and SWSSB. We also use the interface as a probe to characterize the mixed state phases, and specifically, when there is no local modification to preserve the symmetries across the interface, the two sides of the interface are in distinct phases.
Abstract Symmetry-protected topological (SPT) phases are conventionally formulated for pure states protected by exact symmetries. In open quantum systems, however, decoherence generates mixed-state ensembles in which average symmetry can instead emerge only after averaging over microscopic trajectories. In this work, we realize decoherence-induced average SPT (ASPT) order on programmable quantum processors. Starting from a one-dimensional cluster SPT, we engineer sublattice-selective Pauli-ZZZ dephasing through ancilla-assisted quantum circuits. We diagnose the loss of strong symmetry through the decay of the symmetry charge, while the engineered channel preserves the corresponding symmetry at the ensemble level. Moreover, a two-replica R\'enyi-2 string correlator measured through destructive SWAP readout remains nontrivial, revealing the ASPT structure encoded at the density-matrix level. We further show that the engineered dephasing redistributes spectral weight among reduced-stabilizer sectors while preserving the characteristic twofold pairing of the half-chain entanglement spectrum. These results establish a gate-based route to engineering and probing ASPT order and demonstrate structured decoherence as a programmable resource for realizing mixed-state topological quantum matter.