Short-Time Survival and Subsystem-Resolved Transition Rates in Open Quantum Dynamics

This manuscript presents a methodological synthesis of short-time survival probabilities, purity loss, and subsystem-resolved transition rates in finite-dimensional quantum dynamics. Starting from a pure initial state, the analysis relates the initial survival-loss rate to the trace of the established transition-rate operator and to the initial rate of linear-entropy growth. These relations are formulated for general right-differentiable physical state trajectories, without requiring a Lindblad master equation or Markovian dynamics. The GKSL setting supplies an explicit representation in terms of jump operators. For pure product preparations, compatible local projectors resolve the orthogonal state block into nonnegative departure-sector weights. A standard coverage transform connects these weights to subsystem survival slopes and an initial linear-entropy balance. The manuscript also presents a direct measurement procedure, with finite-time bias bounds and statistical uncertainty estimates. Worked examples clarify the limitations of survival-based diagnostics: overlap revival need not indicate non-Markovianity, simultaneous subsystem departures need not generate entanglement, and an entangled conditionally projected state need not imply a nonclassical noise source. The discussion also connects the transition-operator spectrum to boundary entropy asymptotics and distinguishes initial purity loss from the established coherent entanglement-timescale result. Version 9 substantially revises the mathematical assumptions, attribution, and interpretation of earlier LCQI formulations. Established results are explicitly credited. The contribution is their organization, subsystem application, and operational analysis; no new fundamental dynamical law or thermodynamic arrow of time is claimed.