Note on Strong Quantum Markov Properties

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 $\mathit{\text{strong}}\mathit{\text{Markov}}\mathit{\text{property}}$: 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 $\mathit{\text{single}}\mathit{\text{copy}}$ 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.