On the emergence of quantum mechanics from stochastic processes

TL;DR

This paper explores how quantum mechanics can emerge from stochastic processes by generalizing the stochastic-quantum correspondence, linking stochastic kernels to quantum maps, and identifying conditions under which quantum dynamics arise from classical stochastic models.

Contribution

It introduces a generalized framework connecting stochastic kernels with quantum CPTP maps and establishes a divisibility criterion for the emergence of quantum dynamics from stochastic processes.

Findings

Lifts stochastic kernels to quantum maps using a new formalism

Identifies Chapman--Kolmogorov divisibility as key for quantum emergence

Provides a criterion for when stochastic kernels produce quantum dynamics

Abstract

The stochastic--quantum correspondence reinterprets quantum dynamics as arising from an underlying stochastic process on a configuration space. We generalize the correspondence by lifting an arbitrary stochastic kernel $\Gamma$ in finite dimension to a map $\phi$ on $B(\mathcal H)$, formulating the associated lift-compatibility relation, and giving an explicit dictionary between $\Gamma$ and CPTP (Kraus) maps. We isolate Chapman--Kolmogorov divisibility of the lifted family as the decisive additional constraint: when a CK-consistent CPTP family exists, the lift admits a Lindblad master equation form. In this picture, off-diagonal (phase) degrees of freedom act as a compressed carrier of history dependence not fixed by transition kernels alone; conversely, the apparent emergence of quantum phase information from a phase-blind stochastic description is explained as a memory effect. Finally, we state and prove a divisibility criterion for the underlying stochastic kernels, expressed as a condition involving divisibility of the lifted map together with a diagonality requirement on the density operator.