The connection between quantum state recovery and quantum conditional mutual information (QCMI) is studied for the class of purely generated finitely correlated states (pgFCSs) of one-dimensional quantum spin chains. For a tripartition of the chain into two subsystems separated by a buffer region, it is shown that a pgFCS is an approximate quantum Markov chain (QMC) and, stronger, may be approximated by a quantum Markov chain in trace distance, with an error exponentially small in the buffer size. This implies that (1) a locally corrupted state can be approximately recovered by action of a quantum channel on the buffer system, and (2) QCMI is exponentially small in the size of the buffer region. Bounds on the exponential decay rate of QCMI and examples of quantum recovery channels are presented.
We establish improved locally Markovian properties of quantum Gibbs states at every fixed positive temperature. For finite-range interactions of bounded interaction degree, the conditional mutual information (CMI) decays exponentially with separation, with a prefactor exponential in the boundary size of only one of the two regions and independent of the size of the other region and the total system. This improves the exponential volume dependence of the local Markov bounds established by Chen and Rouzé \cite{Chen2025}. For exponentially and stretched-exponentially decaying interactions, we obtain stretched-exponential CMI decay with a boundary-dependent prefactor. For power-law interactions, we prove inverse-logarithmic or algebraic decay with polynomial region-size dependence. Our proof is static, in contrast to the dynamical approach of Chen and Rouzé. We compare the Gibbs state with a product state obtained by removing boundary interactions, reducing the CMI bound to a relative-entropy loss that we control using locality estimates.
A uniform matrix product state defined on a tripartite system of spins, denoted by ABC, is shown to be an approximate quantum Markov chain when the size of subsystem B, denoted |B|, is large enough. The quantum conditional mutual information (QCMI) is investigated and proved to be bounded by a function proportional to exp(−q(|B| − K) + 2K ln |B|), with q and K computable constants. The properties of the bounding function are derived by a new approach, with a corresponding improved value given for its asymptotic decay rate q. We show the improved value of q to be optimal. Numerical investigations of the decay of QCMI are reported for a collection of matrix product states generated by selecting the defining isometry with respect to Haar measure.
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.
Quantum conditional mutual information (QCMI) is a fundamental measure of conditional independence, with central applications throughout quantum information theory and many-body physics. Despite its importance, finding a fully quantum Rényi generalization that satisfies all desirable structural properties has remained an open question. In this work, we introduce a quantum Rényi conditional mutual information possessing these desirable properties: it is monotone under local quantum channels, additive under tensor products, monotone in the Rényi parameter, and converges to the QCMI in the limit
α→1. We establish bounds on this quantity in terms of relative entropies to the Petz-recovered state, showing that it provides a distinct measure of approximate conditional independence. As a consequence, we derive recovery-based upper and lower bounds on the QCMI in terms of quantum generalizations of the Kullback-Leibler divergence, improving upon the best known results in certain regimes. Finally, we establish bounds in terms of relative entropies to the sets of Markov and separable states and derive Rényi chain rules, which appear to be novel even in the classical setting.
Abstract It is well known that the conditional mutual information of a quantum state is zero if, and only if, the quantum state is a quantum Markov chain. Replacing the Umegaki relative entropy in the definition of the conditional mutual information by the Belavkin–Staszewski (BS) relative entropy, we obtain the BS-conditional mutual information, and we call the states with zero BS-conditional mutual information Belavkin–Staszewski quantum Markov chains. In this article, we establish a correspondence which relates quantum Markov chains and BS-quantum Markov chains. This correspondence allows us to find a recovery map for the BS-entropy in the spirit of the Petz recovery map. Furthermore, we show that, over the set of BS-quantum Markov chains, this correspondence constitutes an entanglement-breaking map. Moreover, we prove a structural decomposition of the Belavkin–Staszewski quantum Markov chains and also study states for which the BS-conditional mutual information is only approximately zero. We subsequently extend the aforementioned correspondence, structural decomposition, and recovery map to arbitrary pairs of states and quantum channels. As an application of the correspondence, we find a family of states with non-vanishing conditional mutual information for which it decays superexponentially fast with the size of the middle system.
Quantum many-body Gibbs states satisfy an approximate local Markov property~\cite{chen2025GibbsMarkov}: local noise can be approximately recovered by a quasi-local recovery map, and the conditional mutual information decays for the corresponding tripartition. Recent work~\cite{bergamaschi2025structural} extends this property to approximate stationary states (metastable states) of certain master equations modeling system--bath dynamics, and proposes a strengthened post-selected recovery property requiring recovery to hold for each measurement outcome. In this note, we characterize this \textit{strong Markov property}: it holds if and only if the state additionally satisfies correlation decay for suitable pairs of observables.
We further prove several structural and operational consequences of the strong Markov property in the presence of an underlying master equation. First, one can estimate multiple observables from a \textit{single copy} of the state via a repeated measurement--recovery protocol. Second, any two strongly Markov states must have local marginals that are either very close or well separated. Third, if a strongly Markov state can be expressed as a mixture of two strongly Markov states, then their local marginals must be nearly indistinguishable.