Stability of mixed-state quantum phases via finite Markov length

Abstract

For quantum phases of Hamiltonian ground states, the energy gap plays a central role in ensuring the stability of the phase as long as the gap remains finite. We propose Markov length, the length scale at which the quantum conditional mutual information (CMI) decays exponentially, as an equally essential quantity characterizing mixed-state phases and transitions. For a state evolving under a local Lindbladian, we argue that if its Markov length remains finite along the evolution, then it remains in the same phase, meaning there exists another quasi-local Lindbladian evolution that can reverse the former one. We apply this diagnostic to toric code subject to decoherence and show that the Markov length is finite everywhere except at its decodability transition, at which it diverges. CMI in this case can be mapped to the free energy cost of point defects in the random bond Ising model. This implies that the mixed state phase transition coincides with the decodability transition and also suggests a quasi-local decoding channel.

5 + 7 pages, 3 main figures; v2: simpler derivation of quasi-local decoder

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• Locally Purified Density Operators for Symmetry-Protected Topological Phases in Mixed States
• Efficient Detection of Strong-To-Weak Spontaneous Symmetry Breaking via the Rényi-1 Correlator
• Holographic View of Mixed-State Symmetry-Protected Topological Phases in Open Quantum Systems
• Spontaneous symmetry breaking in open quantum systems: strong, weak, and strong-to-weak
• Mixed-state phase transitions in spin-Holstein models
• Universal Characterization of Quantum Many-Body States through Local Information
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• Gauge theory and mixed state criticality
• Mixed-State Topological Order under Coherent Noise
• Fundamental thresholds for computational and erasure errors via the coherent information
• Stabilizing Non-Abelian Topological Order against Heralded Noise via Local Lindbladian Dynamics
• A New Framework for Quantum Phases in Open Systems: Steady State of Imaginary-Time Lindbladian Evolution
• Exactly solvable dissipative dynamics and one-form strong-to-weak spontaneous symmetry breaking in interacting two-dimensional spin systems
• The surface code beyond Pauli channels: Logical noise coherence, information-theoretic measures, and errorfield-double phenomenology
• Instability of steady-state mixed-state symmetry-protected topological order to strong-to-weak spontaneous symmetry breaking
• Robust Mixed-State Cluster States and Spurious Topological Entanglement Negativity
• Spectral properties and coding transitions of Haar-random quantum codes
• Learning mixed quantum states in large-scale experiments
• Non-Commutative weak measurements: Entanglement, Symmetry Breaking, and the Role of Readout
• Universal Decay of Mutual Information and Conditional Mutual Information in Gapped Pure- and Mixed-State Quantum Matter
• Noisy Monitored Quantum Circuits
• Holographic duality between bulk topological order and boundary mixed-state order
• Spacetime Markov length: a diagnostic for fault tolerance via mixed-state phases
• Phases of Floquet code under local decoherence