Theorem: Classical Markov boundary of the quantum network

If $H_M=0$, the node populations $p_i={\rho }_{ii}$ obey, for every initial density operator,

They do not depend on initial node coherence. Diagonal inputs remain diagonal. Node dephasing has no effect on these population trajectories.

With diagonal preparations, node-basis readouts, and interventions specified by nonnegative substochastic matrices whose complete sum is stochastic, every allowed record history has an exact classical continuous-time Markov realization.

Proof. For $L=|i⟩⟨j|$, the gain term contributes $p_j$ to population $i$. Its anticommutator removes $p_j$ from population $j$. Summing gives (eq:markov). The node-projector dissipators have zero diagonal contribution. At a diagonal state they generate no off-diagonal terms.

An intervention matrix $A_r$ acts on diagonal populations by $p\mapsto A_{r}p$. Its branch probability is the sum of the resulting entries. Between interventions, the Markov propagator acts by a stochastic matrix. Products of these matrices give exactly the same unnormalized history populations and record probabilities as the quantum maps. Induction proves the complete-history claim.