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.
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.
Statistical mechanics assumes that a quantum many-body system at low temperature can be effectively described by its Gibbs state. However, many complex quantum systems exist only as metastable states of dissipative open system dynamics, which appear stable and robust yet deviate substantially from true thermal equilibrium. In this work, we model metastable states as approximate stationary states of a quasi-local, (KMS)-detailed-balanced master equation representing Markovian system-bath interaction, and unveil a universal structural theory: all metastable states satisfy an area law of mutual information and a Markov property. The more metastable the states are, the larger the regions to which these structural results apply. Therefore, the hallmark correlation structure and noise resilience of Gibbs states are not exclusive to true equilibrium but emerge dynamically. Behind our structural results lies a systematic framework encompassing sharp equivalences between local minima of free energy, a non-commutative Fisher information, and approximate detailed balance conditions. Our results build towards a comprehensive theory of thermal metastability and, in turn, formulate a well-defined, feasible, and repeatable target for quantum thermal simulation.
We propose a unifying framework for characterizing pure and mixed state phases of matter across equilibrium, non equilibrium, and metastable regimes. We introduce the concept of locally stable states, defined by the operational property that any local operation (including post selection) can be reversed by a local channel. We prove that local stability is equivalent to a state being short range correlated, defined by the decay of both correlations and conditional mutual information. We demonstrate that these properties are invariant under locally reversible channels, thus defining locally stable phases. Furthermore, we prove that local stability implies both the decay of a family of nonlinear correlators, including the fidelity correlator, and the decay of correlations in the canonical purification, thus bridging the gap between mixed and pure states. Along the way, we establish two results which may be of independent interest: we show that post-selection on locally stable (short range correlated) states can be implemented via local channels and that quantum Markov chains can be characterized by the local computability of nonlinear observables.
The Markov property entails the conditional independence structure inherent in Gibbs distributions for general classical Hamiltonians, a feature that plays a crucial role in inference, mixing time analysis, and algorithm design. However, much less is known about quantum Gibbs states. In this work, we show that for any Hamiltonian with a bounded interaction degree, the quantum Gibbs state is locally Markov at arbitrary temperature, meaning there exists a quasi-local recovery map for every local region. Notably, this recovery map is obtained by applying a detailed-balanced Lindbladian with jumps acting on the region. Consequently, we prove that (i) the conditional mutual information (CMI) for a shielded small region decays exponentially with the shielding distance, and (ii) under the assumption of uniform clustering of correlations, Gibbs states of general non-commuting Hamiltonians on D-dimensional lattices can be prepared by a quantum circuit of depth eO(logD(n/ε)), which can be further reduced assuming certain local gap condition. Our proofs introduce a regularization scheme for imaginary-time-evolved operators at arbitrarily low temperatures and reveal a connection between the Dirichlet form, a dynamic quantity, and the commutator in the KMS inner product, a static quantity. We believe these tools pave the way for tackling further challenges in quantum thermodynamics and mixing times, particularly in low-temperature regimes.
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.
Given a quantum channel and a state which satisfy a fixed point equation approximately (say, up to an error ε), can one find a new channel and a state, which are respectively close to the original ones, such that they satisfy an exact fixed point equation? It is interesting to ask this question for different choices of constraints on the structures of the original channel and state, and requiring that these are also satisfied by the new channel and state. We affirmatively answer the above question, under fairly general assumptions on these structures, through a compactness argument. Additionally, for channels and states satisfying certain specific structures, we find explicit upper bounds on the distances between the pairs of channels (and states) in question. When these distances decay quickly (in a particular, desirable manner) as ε→0, we say that the original approximate fixed point equation is rapidly fixable. We establish rapid fixability, not only for general quantum channels, but also when the original and new channels are both required to be unitary, mixed unitary or unital. In contrast, for the case of bipartite quantum systems with channels acting trivially on one subsystem, we prove that approximate fixed point equations are not rapidly fixable. In this case, the distance to the closest channel (and state) which satisfy an exact fixed point equation can depend on the dimension of the quantum system in an undesirable way. We apply our results on approximate fixed point equations to the question of robustness of quantum Markov chains (QMC) and establish the following: For any tripartite quantum state, there exists a dimension-dependent upper bound on its distance to the set of QMCs, which decays to zero as the conditional mutual information of the state vanishes.
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.