On the emergence of quantum mechanics from stochastic processes

The stochastic--quantum correspondence reinterprets quantum dynamics in terms of an underlying stochastic process on a configuration space. We generalize its lifting prescription, overcoming restrictions on the reachable density matrices without enlarging the configuration space or its associated Hilbert space. A compatibility relation and Kraus dictionary connect stochastic kernels to completely positive, trace-preserving (CPTP) maps: every finite-dimensional kernel admits a compatible lift, and these lifts collectively encompass all quantum channels as the kernels vary. We describe sufficiently regular composable CPTP dynamics through time-local Lindblad generators. We establish that every rooted quantum dynamical family admits a stochastic realizer on the same configuration space, and that a revival of configuration-space distinguishability rules out every Markovian realizer on that space. We show how the predictive information carried by off-diagonal degrees of freedom in a compatible composable lift is encoded in the continuation probabilities of a family of context-dependent stochastic processes. We also analyze physical measurement by separating active basis rotations from configuration recording. The rooted quantum probabilities alone do not enforce faithfulness in idealized recording interactions; nevertheless, we show that a faithful stochastic realizer always exists.