Recent progress in open many-body quantum systems has highlighted the importance of the Markov length, the characteristic scale over which conditional correlations decay. It has been proposed that non-equilibrium phases of matter can be defined as equivalence classes of states connected by short-time evolution while maintaining a finite Markov length, a notion called local reversibility. A natural question is whether well-known classical models of non-equilibrium criticality fit within this framework. Here we investigate the Domany-Kinzel model -- which exhibits an active phase and an absorbing phase separated by a 1+1-D directed-percolation transition -- from this information-theoretic perspective. Using tensor network simulations, we provide evidence for local reversibility within the active phase. Notably, the Markov length diverges upon approaching the critical point, unlike classical equilibrium transitions where Markov length is zero due to their Gibbs character. Correspondingly, the conditional mutual information exhibits scaling consistent with directed percolation universality. Further, we analytically study the case of 1+1-D compact directed percolation, where the Markov length diverges throughout the phase diagram due to spontaneous breaking of domain-wall parity symmetry from strong to weak. Nevertheless, the conditional mutual information continues to faithfully detect the corresponding phase transition.
Conditional mutual information (CMI) characterizes dependence between two regions conditioned on an intervening buffer. Its characteristic decay length, the Markov length, has recently been identified as a diverging length scale at mixed-state phase transitions. Here, we investigate CMI dynamics in one-dimensional classical stochastic models and propose a universal scaling form for systems whose Markov length diverges with time. We derive this form in the inactive phase of parity-conserving directed percolation, where domain-wall coarsening generates a growing length scale and a self-similar probability distribution. We introduce a Hankel-based truncation scheme that enables matrix-product-state simulations at long times and over large spatial regions. Using this method, we verify the proposed scaling both in the inactive phase and at the nonequilibrium critical point, where the dynamical critical exponent governs Markov-length growth. More broadly, the same scaling form captures cases in which collective inference sets the diverging Markov length. Despite these distinct mechanisms, the CMI between fixed microscopic regions exhibits a common scaling dimension of two. Finally, we extend this framework to temporal CMI, establishing an analogous relation between the strength of conditional dependence in time domain and its characteristic memory time.
Conditional mutual information (CMI) probes spatial non-Markovianity and plays a central role in a recently developed framework for defining mixed-state phases based on local reversibility. This framework sharply distinguishes exponentially decaying from algebraically decaying CMI, since the latter obstructs equivalence to a state with finite Markov length. Here we show that states that flow to the same renormalization-group (RG) fixed point can nevertheless exhibit qualitatively different CMI. Starting from a critical Gibbs state of a finite-range local classical Hamiltonian, whose CMI vanishes beyond the interaction range, we show that an RG-irrelevant perturbation that breaks detailed balance generically generates algebraically decaying CMI. Crucially, these power-law tails are cutoff-suppressed: although they lead to an infinite Markov length at every fixed lattice cutoff, they vanish as a positive power of the cutoff in the continuum limit. We construct solvable models that exhibit this phenomenon and predict that the Ising-symmetric critical point of Toom's cellular automata generically exhibits such behavior. We also find analogous cutoff-suppressed CMI in the high-temperature paramagnetic phase of the long-range Ising paramagnet, in contrast with the genuine power-law CMI at the finite-temperature critical point in this same model that survives the continuum limit. We also derive a general result that in one spatial dimension, CMI decays faster than the inverse square of the buffer length if and only if the long-distance physics is Markovian.
Changes in the entanglement structure and critical phenomena are hallmarks of quantum phase transitions. Here, we discuss how they appear in transitions between classes of states with distinct entanglement patterns beyond the paradigm of stable equilibrium phases of matter. Using a mapping between stochastic automata and isometric Tensor Network States (isoTNS), we construct a two-dimensional quantum state from the Domany-Kinzel automaton, which is a (1+1)D process with an absorbing phase transition in the directed percolation class. At the critical point of the automaton, the corresponding isoTNS hosts algebraic correlations in all spatial directions. The continuous parent Hamiltonian of this state has a degenerate ground state manifold. It consists of a product state (the absorbing state) and a second state that undergoes a transition from pairwise entanglement between distant regions, similarly to the W state, to a state with trivial entanglement. Our results demonstrate how the correspondence between isoTNS and classical stochastic evolution can be used to probe the Hilbert space structure beyond stable ground state manifolds.
Quantum critical phases are extended regions of phase space characterized by a diverging correlation length. By analogy, we define an \emph{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 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
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 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 toric code and show that its logical error rate saturates to a fractional value 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.
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.
Under local Lindbladian evolution, the growth of conditional mutual information and its associated Markov length scale is not limited by Lieb-Robinson bounds, yet it has been unclear how quickly the Markov length can increase. We show that an arbitrarily short, strictly local Lindbladian can generate infinite Markov length from a product state. We also construct strictly local Lindbladians that converge rapidly to steady states with algebraically decaying conditional mutual information. Despite their long-range conditional correlations, the states we construct are local reversibly connected to a product state by sublinear depth circuits. Thus infinite Markov length does not imply slow relaxation nor obstruct phase stability when suitably defined.