⚛️ quantum physics

Fate of average symmetry-protected topological states under symmetry-preserving quantum operations

This paper investigates the robustness of average symmetry-protected topological (ASPT) states under symmetry-preserving quantum operations, revealing that while these states remain stable across a broad range of decoherence and postselection strengths, they ultimately undergo strong-to-weak or weak-symmetry spontaneous symmetry breaking in the maximal-decoherence and projective limits, respectively.

Original authors: Yoshihito Kuno, Takahiro Orito

Published 2026-09-24

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Yoshihito Kuno, Takahiro Orito

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, particles can become linked in ways that defy our everyday intuition, forming patterns of order that are invisible to the naked eye but fundamental to the nature of matter. Scientists have long studied a special class of these patterns called symmetry-protected topological states. Think of these as intricate, knotted structures that remain stable only as long as the system is perfectly isolated and follows specific rules of balance, known as symmetries. However, the real world is rarely perfect. Quantum systems are constantly bombarded by their environment, leading to a loss of information called decoherence. When this happens, the strict rules that protect the quantum order can weaken, turning a "strong" symmetry into a "weak" one. This creates a new, messy kind of quantum state known as an average symmetry-protected topological state. The big question for physicists is whether these fragile, mixed-up states can survive further disturbances, or if they will collapse into something entirely different.

A team of researchers in Japan has taken a deep dive into this question, focusing on a specific one-dimensional quantum model that acts like a chain of linked spins. They wanted to see what happens when this quantum chain is subjected to two different types of symmetry-preserving operations: one that introduces random noise (decoherence) and another that involves a careful selection process based on measurement outcomes (postselection). To understand these complex states, the researchers used a mathematical trick that allows them to view the messy, mixed quantum state as if it were a clean, pure state existing in a doubled space. By introducing two effective spins on each rung of this doubled ladder, they created a clear map of the system's internal structure. This map revealed exactly which parts of the quantum order were held together by unbreakable rules and which parts were free to change.

The researchers found that the quantum order is far more resilient than one might expect. When they applied a noisy channel that scrambled the spins, the special topological order did not immediately vanish. Instead, it remained robust across a wide range of noise strengths. The system only began to lose its unique character when the noise became overwhelmingly strong, approaching a theoretical maximum. In this extreme limit, the system underwent a precise transformation, shedding its topological protection and settling into a new state where a different kind of symmetry breaking became exact. Crucially, the team showed that this change is not a sudden jump or a sharp phase transition that happens at a specific, moderate level of noise. Rather, the system gradually deforms, with the new order only becoming fully established at the very edge of the extreme limit.

They reached similar conclusions when they tested the system with a different operation: a process where they kept only the results of measurements that did not trigger a specific alarm, a technique known as no-click postselection. This method is often used to purify quantum states, but here it acted as a filter that competed with the existing quantum order. Again, the researchers discovered that the topological state held firm against moderate levels of this filtering. The system did not abruptly switch to a new phase as the filtering strength increased. Instead, the old order persisted through a broad range of conditions, only giving way to a new, symmetry-breaking pattern when the filtering became so intense that it effectively forced the system into a single, rigid configuration.

To reach these conclusions, the team combined theoretical analysis with powerful computer simulations. They built a detailed model of the quantum chain and watched how it evolved under these different operations. Their calculations showed that while simple theories might predict a sudden collapse of the order at a certain point, the actual behavior is much more subtle. The system resists change through a long, gradual crossover, where the characteristic length of the correlations grows rapidly but does not signal a true phase transition until the very end. This means that for a vast range of real-world conditions, these quantum states can maintain their unique properties even as they are disturbed. The study provides a clear picture of how quantum order survives in a noisy world, showing that it is not easily destroyed but rather slowly reshaped until it reaches a breaking point only under the most extreme conditions.

Technical Summary: Fate of Average Symmetry-Protected Topological States under Symmetry-Preserving Quantum Operations

Problem Statement
The characterization of quantum phases in open systems has extended the concept of symmetry-protected topological (SPT) phases to mixed states, specifically average symmetry-protected topological (ASPT) phases. In these states, decoherence reduces strong symmetries to weak symmetries, yet the system may retain nonlocal information inherited from the parent pure SPT state. A critical open question is the robustness of ASPT order against further symmetry-preserving quantum operations (such as additional decoherence channels or postselection) and the nature of the competing orders that emerge when ASPT order is suppressed. While previous studies suggested one-dimensional ASPT states are robust against certain operations, a systematic understanding of this robustness and the specific mechanisms of order replacement remains lacking.

Methodology
The authors investigate the fate of the one-dimensional cluster SPT state under symmetry-preserving operations using the Choi mapping, which represents the mixed state as a pure state in a doubled Hilbert space ( $H_u \otimes H_\ell$ ). The core methodological innovation is the introduction of a $\tau$ - $\eta$ spin mapping on each doubled rung. This mapping transforms the two physical spins on each rung into two effective spin-1/2 degrees of freedom ( $\tau$ and $\eta$ ), rendering the ASPT parent Hamiltonian transparent.

The study employs a multi-pronged approach:

Analytical Construction: The authors derive the exact parent Hamiltonian for the ASPT state and its deformed versions under two distinct operations: (i) an even-site $X$ -decoherence channel and (ii) symmetry-preserving no-click postselection (generated by weak measurements).
Qualitative vs. Exact Analysis: They compare qualitative effective Hamiltonians (often resembling Transverse-Field Ising Models or mean-field Ising models) with the exact parent Hamiltonians derived via transformation schemes. This comparison highlights the role of additional correlated interactions generated by the filtering operations.
Numerical Simulation: Using Density Matrix Renormalization Group (DMRG) and Matrix Product State (MPS) filtering calculations, the authors compute observables such as purity-weighted string correlators, Rényi-2 string correlators, and Operator-Space Entanglement Entropy (OSEE) across varying strengths of decoherence and postselection.

Key Contributions and Results

Unified Microscopic Description via $\tau$ - $\eta$ Mapping:
The $\tau$ - $\eta$ mapping provides a unified framework for characterizing ASPT order. It identifies exact local conserved stabilizers ( $C^\tau_{2m+1}$ and $C^\eta_j$ ) that correspond to specific string orders. Specifically, the mapping reveals that the ASPT state possesses exact Rényi-2 string orders (associated with both upper and lower copies and their combinations) but lacks an exact purity-weighted even-sublattice string correlator. This clarifies the bulk order structure of the cluster-state ASPT.
Fate under Even-Site $X$ -Decoherence (SWSSB Competition):
When subjected to an even-site $X$ -decoherence channel, the ASPT competes with Strong-to-Weak Spontaneous Symmetry Breaking (SWSSB).
- Qualitative View: An effective Hamiltonian suggests a Transverse-Field Ising Model (TFIM) sector, implying a potential phase transition between ASPT string order and SWSSB.
- Exact & Numerical View: The exact parent Hamiltonian contains additional correlated interactions beyond the simple TFIM structure. Numerical results show that the ASPT remains robust over a broad range of decoherence strengths. The transition to exact SWSSB order (characterized by long-range Rényi-2 $ZZ$ correlations) occurs only in the maximal-decoherence limit ( $p_x \to 1/2$ ). For finite strengths, the system exhibits a strong finite-size crossover rather than a sharp phase transition at a critical finite value.
Fate under No-Click Postselection (WSSB Competition):
Under symmetry-preserving no-click postselection, the ASPT competes with Weak-Symmetry Spontaneous Symmetry Breaking (WSSB).
- Qualitative View: A mean-field treatment of the effective Hamiltonian suggests an Ising-like instability toward a WSSB state.
- Exact & Numerical View: The exact parent Hamiltonian expansion reveals that additional correlated interactions appear at the same leading order as the mean-field perturbation. Consequently, the mean-field instability does not guarantee a phase transition at finite postselection strength. Numerical data confirms that the ASPT is robust against finite-strength postselection. Exact WSSB order (characterized by purity-weighted $ZZ$ long-range order) is realized only in the projective limit ( $q_x \to 1$ ).
Robustness and Crossover Behavior:
In both scenarios, the ASPT order is not immediately destroyed by symmetry-preserving nonunitary operations. Instead, the system undergoes a continuous deformation where symmetry-breaking correlations are enhanced only near the respective limiting cases (maximal decoherence or projective measurement). The numerical analysis of system-size dependence indicates that the apparent sharpness of the transition is due to rapidly growing characteristic lengths as the limits are approached, rather than a true critical point at finite operation strength.

Significance
The paper establishes a systematic picture of the robustness of one-dimensional ASPT order. By utilizing the $\tau$ - $\eta$ spin mapping, the authors provide a transparent microscopic link between conserved stabilizers and various string orders. The work demonstrates that while qualitative effective Hamiltonians (like TFIM or mean-field Ising models) correctly identify the competing orders, they fail to predict the critical behavior of the actual filtered states due to the generation of higher-order correlated interactions in the exact parent Hamiltonian. The primary finding is that ASPT order is remarkably robust, persisting over a wide parameter regime, with exact symmetry-breaking orders emerging only asymptotically at the limits of maximal decoherence or projective postselection. This challenges the notion of a simple finite-strength phase transition and highlights the necessity of exact parent Hamiltonian analysis and finite-size scaling in characterizing open quantum phases.

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