Instability of steady-state mixed-state symmetry-protected topological order to strong-to-weak spontaneous symmetry breaking
1Joint Quantum Institute, NIST/University of Maryland, College Park, MD, 20742, USA
2Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, MD, 20742, USA
3Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742, USA
4Google Quantum AI, California, USA
| Published: | 2025-11-17, volume 9, page 1912 |
| Editor: | Angelo Carollo |
| Eprint: | arXiv:2410.12900v2 |
| Doi: | https://doi.org/10.22331/q-2025-11-17-1912 |
| Citation: | Quantum 9, 1912 (2025). |
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Abstract
Recent experimental progress in controlling open quantum systems enables the pursuit of mixed-state nonequilibrium quantum phases. We investigate whether open quantum systems hosting mixed-state symmetry-protected topological states as steady states retain this property under symmetric perturbations. Focusing on the $\textit{decohered cluster state}$ – a mixed-state symmetry-protected topological state protected by a combined strong and weak symmetry – we construct a parent Lindbladian that hosts it as a steady state. This Lindbladian can be mapped onto exactly solvable reaction-diffusion dynamics, even in the presence of certain perturbations, allowing us to solve the parent Lindbladian in detail and reveal previously-unknown steady states. Using both analytical and numerical methods, we find that typical symmetric perturbations cause strong-to-weak spontaneous symmetry breaking at arbitrarily small perturbations, destabilize the steady-state mixed-state symmetry-protected topological order. However, when perturbations introduce only weak symmetry defects, the steady-state mixed-state symmetry-protected topological order remains stable. Additionally, we construct a quantum channel which replicates the essential physics of the Lindbladian and can be efficiently simulated using only Clifford gates, Pauli measurements, and feedback.
Featured image: We study the steady-state phase diagram of a Lindbladian $\mathcal{L}_\lambda$ which interpolates between $\mathcal{L}_\mathcal{C}$, whose steady states exhibit mixed SPT order and $\tilde{\mathcal{L}}_\mathcal{C}$, whose steady states are trivially ordered. Rather than a finite phase of either order, we find that the entire intermediate region exhibits SW-SSB.
Popular summary
Recently, there has been growing interest in mixed-state symmetry-protected topological (SPT) phases which are the nonequilibrium counterparts of well-known pure-state SPT phases. In the pure-state setting, if we apply small symmetric perturbations to a Hamiltonian whose ground states are SPTs, the ground states of the perturbed Hamiltonian will remain in the same SPT phase. We ask the analogous question for open quantum systems, which are described by Lindbladians rather than Hamiltonians: If we apply small symmetric perturbations to a Lindbladian whose steady states are mixed-state SPTs, will the steady states of the perturbed Lindbladian remain in the same mixed-state SPT phase?
To explore this, we study a particular mixed-state SPT known as the decohered cluster state. We construct a Lindbladian whose steady state realizes this decohered cluster, then introduce symmetric perturbations and analyze the resulting steady states.
Our main finding is that except for a special class of perturbations, even infinitesimal symmetric perturbations cause the system to undergo a novel phenomenon known as strong-to-weak spontaneous symmetry breaking, which destabilizes the mixed-state SPT order. We establish this numerically as well as analytically for a broad class of perturbations.
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