The Markov length, the length scale governing the exponential decay of conditional mutual information (CMI) in a quantum many-body state, has proven useful for characterizing mixed-state phases of matter. Nevertheless, we show that a state's Markov length is not generally stable under weak local channels. The failure of stability occurs already at a pure product state. We demonstrate this explicitly in a one-dimensional fermionic chain, where a fine-tuned depth-two circuit of nearest-neighbor Gaussian channels maps the vacuum to a state whose CMI decays \emph{algebraically} and whose Markov length is therefore infinite, even when the channels are taken arbitrarily close to the identity. The algebraic decay is rooted in the nonanalyticity of the von Neumann entropy at rank-deficient density operators. We exhibit an analogous instability on an ordinary qubit chain. More generally, we show that the Markov length is unstable in any translation-invariant state with even a single rank-deficient reduced density operator. These results pose a challenge to recent definitions of mixed-state phase equivalence based on the Markov length.
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.
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.
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.
Quantum low-density parity-check (qLDPC) codes have a canonical mixed state---the normalized projector onto the code space---that lies at the intersection of quantum error correction and quantum many-body physics, yet the stability of its information-theoretic structure under local noise remains poorly understood. Recent progress on mixed-state phases of matter has identified the spatial decay of conditional mutual information (CMI)
I(A:C∣B) as a key diagnostic of such stability, analogous to the role of the spectral gap for quantum ground states. We prove that, for the normalized code-projector state of a
Zd qLDPC stabilizer code under local Pauli noise, the CMI across a buffer
B separating
A from
C decays exponentially with the width of
B in two complementary noise regimes independent of system size. At weak noise, this holds for full-system tripartitions, where correlations between the two sides require an error cluster to cross the buffer, an exponentially unlikely event. At strong noise, exponential decay holds whenever the buffer supports no logical operator, since correlations can persist only along rare chains of buffer qudits that remain informative. Under an additional space-uniformity assumption, we establish the robustness of Levin--Wen topological entropy. On finite-dimensional lattices of size
n, the weak-noise bound further implies a constant-layer quasi-local Petz recovery channel of range $O(\poly(\log n)/\varepsilon)$ that recovers the noiseless state within trace-distance error
ε. Both CMI bounds apply to general qLDPC interaction graphs, including geometries where such constant-depth quasi-local recovery need not exist. Thus, the exponential decay of the noise-induced CMI provides a general criterion for mixed-state phase stability.
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.
A departure of a mixed quantum state from a local Gibbs description is generally invisible to local observables but can be detected by the conditional mutual information (CMI). Here we study the relationship between the renormalization group (RG) and CMI, and in particular, how RG constrains CMI. We first show that the CMI between nonadjacent regions
A and
C, conditioned on the buffer region
B, is UV-finite whenever the state admits a locally reversible RG with a fixed on-site Hilbert space dimension. We then study two hierarchical models that have long-range CMI and yet admit a simple RG description. The first model has a divergent Markov length at every temperature
0<T<∞ but nevertheless flows to an infinite-temperature product state under RG. The second model satisfies the local Markov condition while violating the global one and is stable against weak noise. At the critical noise strength, the two-point CMI decays only polynomially as a function of the system size, while the two-point mutual information vanishes.