Ensembles of Propagators, Not Single Hamiltonians, Govern Open Quantum Dynamics

We recast open quantum dynamics from the conventional single-Hamiltonian picture—which presumes factorized initial states—into an \emph{ensemble} description: interacting subsystems are most naturally represented by an ensemble of two-time propagators \(\{\Phi_{S_i}(t_1,t_2)\}\) that remain valid for arbitrary system--environment correlations. Working with an exactly solvable \(N\)-qubit network that conserves excitation number, we derive closed-form two-time propagators and show that positivity and complete positivity (CP) of \(\Phi(t_1,t_2)\) are governed not by the mere presence of initial correlations but by their \emph{change} across \((t_1,t_2)\). Growing correlations yield positive (and CP) propagators, whereas decaying correlations yield nonpositive (and non-CP) propagators. Consequently, monitoring positivity across the ensemble \(\{\Phi_{S_i}(t_1,t_2)\}\) provides an operational probe of otherwise inaccessible environmental structure.