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arXiv:2512.22121

Information Critical Phases under Decoherence

Akash Vijay, Jong Yeon Lee

quant-phcond-mat.dis-nncond-mat.stat-mechSubmitted Dec 26, 2025• Announced Dec 29, 2025• 0 citations
v1 · Dec 26, 2025v2 · Aug 10, 2026
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Abstract

Quantum critical phases are extended regions of phase space characterized by a diverging correlation length. By analogy, we define an information critical phase as an extended region of a mixed state phase diagram where the Markov length, the characteristic length scale governing the decay of the conditional mutual information (CMI), diverges. We demonstrate that such a phase arises in decohered $\mathbb{Z}_{N}$ Toric codes by assessing both the CMI and the coherent information, the latter quantifying the robustness of the encoded logical qudits. For $N>4$, we find that the system hosts an information critical phase intervening between the decodable and non-decodable phases where the coherent information saturates to a fractional value in the thermodynamic limit, indicating that a finite fraction of logical information is still preserved. We show that the density matrix in this phase can be decomposed into a convex sum of Coulombic pure states, where gapped anyons reorganize into gapless photons. We further consider the ungauged $\mathbb{Z}_{N}$ Toric code and interpret its mixed state phase diagram in the language of strong-to-weak spontaneous symmetry breaking. We argue that in the dual model, the information critical phase arises because the spontaneously broken off-diagonal $\mathbb{Z}_{N}$ symmetry gets enhanced to a U(1) symmetry, resulting in a novel superfluid phase whose gapless modes involve coherent excitations of both the system and the environment. Finally, we propose an optimal decoding protocol for the corrupted $\mathbb{Z}_{N}$ Toric code and evaluate its effectiveness in recovering the fractional logical information preserved in the information critical phase. Our findings identify a gapless analog for mixed-state phases that still acts as a fractional topological quantum memory, thereby extending the conventional paradigm of quantum memory phases.

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Quantum critical phases are extended regions of phase space characterized by a diverging correlation length. By analogy, we define an information critical phase as an extended region of a mixed state phase diagram where the Markov length, the characteristic length scale governing the decay of the conditional mutual information (CMI), diverges.
We demonstrate that such a phase arises in decohered ZN\mathbb{Z}_{N}ZN​ Toric codes by assessing both the CMI and the coherent information, the latter quantifying the robustness of the encoded logical qudits. For N>4N>4N>4, we find that the system hosts an information critical phase intervening between the decodable and non-decodable phases where the coherent information saturates to a fractional value in the thermodynamic limit, indicating that a finite fraction of logical information is still preserved. We show that the density matrix in this phase can be decomposed into a convex sum of Coulombic pure states, where gapped anyons reorganize into gapless photons. We further consider the ungauged ZN\mathbb{Z}_{N}ZN​ Toric code and interpret its mixed state phase diagram in the language of strong-to-weak spontaneous symmetry breaking. We argue that in the dual model, the information critical phase arises because the spontaneously broken off-diagonal ZN\mathbb{Z}_{N}ZN​ symmetry gets enhanced to a U(1) symmetry, resulting in a novel superfluid phase whose gapless modes involve coherent excitations of both the system and the environment. Finally, we propose an optimal decoding protocol for the corrupted ZN\mathbb{Z}_{N}ZN​ Toric code and evaluate its effectiveness in recovering the fractional logical information preserved in the information critical phase. Our findings identify a gapless analog for mixed-state phases that still acts as a fractional topological quantum memory, thereby extending the conventional paradigm of quantum memory phases.
We demonstrate the existence of an extended non-equilibrium critical phase, characterized by sub-exponential decay of conditional mutual information (CMI), in the surface code subject to heralded random Pauli measurement channels. By mapping the resulting mixed state to the ensemble of completely packed loops on a square lattice, we relate the extended phase to the Goldstone phase of the loop model. In particular, CMI is controlled by the characteristic length scale of loops, and we use analytic results of the latter to establish polylogarithmic decay of CMI in the critical phase. We find that the critical phase retains partial logical information that can be recovered by a global decoder, but not by any quasi-local decoder. To demonstrate this, we introduce a diagnostic called punctured coherent information which provides a necessary condition for quasi-local decoding.
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 quasilocal 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 quasilocal decoding channel.
Quantum error correction protects quantum information against decoherence, provided the noise strength remains below a critical threshold. This threshold marks the critical point for the decoding phase transition. Here, we connect this transition in the toric code to a topological phase transition in disordered Majorana fermions at high temperatures. A quantum memory in the error-correctable phase is captured by the presence of a Majorana zero mode, trapped in vortex defects associated with twisted boundary conditions. These results are established by expressing the coherent information, which measures the amount of recoverable quantum information in a given noisy code, in terms of a mixed-state topological order parameter of fermions. Our work hints at a broader connection of the robustness of quantum information in stabilizer codes and mixed-state topological phase transitions in symmetry-protected fermion matter.
Quantum error correction protects quantum information against decoherence provided the noise strength remains below a critical threshold. This threshold marks the critical point for the decoding phase transition. Here we connect this transition in the toric code to a topological phase transition in disordered Majorana fermions at high temperatures. A quantum memory in the error correctable phase is captured by the presence of a Majorana zero mode, trapped in vortex defects associated with twisted boundary conditions. These results are established by expressing the coherent information, which measures the amount of recoverable quantum information in a given noisy code, in terms of a mixed-state topological order parameter of fermions. Our work hints at a broader connection of the robustness of quantum information in stabilizer codes and mixed-state topological phase transitions in symmetry protected fermion matter.
A topological phase can undergo a phase transition driven by anyon condensation. A potential obstruction to such a mechanism could arise if there exists a symmetry between anyons that have non-trivial mutual statistics. Here we consider toric code subjected to errors that tend to proliferate anyons with non-trivial mutual statistics. Using triangle inequality, we show that in the presence of electromagnetic duality and a partial-transpose symmetry, a decoherence induced phase transition out of the topological phase must be rather unconventional and lie beyond standard rules of anyon condensation. To explore such physics, we first subject toric code to a self-dual quantum channel where Kraus operators are proportional to X+Z. We find that the topological phase is stable up to the maximal error rate, when viewing density matrix as a pure state in the double Hilbert space. To access an unconventional transition, we then consider a perturbed toric code subjected to the self-dual channel, and find numerical evidence that beyond a critical error rate, the topological phase is destroyed resulting in a critical phase where anyons are only power-law condensed.
Coherent noise can give rise to unconventional regimes of quantum information recovery and mixed-state order. We study these phenomena in two settings related by Wegner duality: the triangular-lattice surface code subject to single-qubit Pauli-ZZZ rotations followed by syndrome measurements, and a Z2\mathbb{Z}_2Z2​ paramagnet subject to strongly symmetric decoherence. Increasing the rotation angle drives a transition from a decodable phase to a \emph{partially decodable} critical phase in the code, and from a strongly symmetric phase to a phase with \emph{algebraic} Z2\mathbb{Z}_2Z2​ strong-to-weak spontaneous symmetry breaking (SWSSB) in the dual paramagnet. We develop a common description of these phenomena in terms of a weakly coupled SO(n)\mathrm{SO}(n)SO(n) nonlinear sigma model in the replica limit n→1n\to1n→1. Taking this replica limit analytically, we derive universal signatures of the critical phase: an optimal Pauli decoding fidelity that approaches a universal function of the system's aspect ratio in the thermodynamic limit, and a dual fidelity correlator that decays as a universal power law. These observables relate to correlations of twist operators in the sigma model description. Large-scale numerical simulations confirm these analytical predictions. We also argue that the algebraic SWSSB phase cannot be realized by any Pauli dephasing channel acting on the Z2\mathbb{Z}_{2}Z2​ paramagnet. We probe this "non-classical" character through the output state's sensitivity to well-separated Z2\mathbb{Z}_{2}Z2​ fluxes inserted in the initial state, which reveals persistent coherence between distinct Z2\mathbb{Z}_{2}Z2​ charge trajectories at long distances.