# Classical quantum stochastic processes

## TL;DR

This paper explores the relationship between quantum coherence, Markovianity, and classicality in quantum stochastic processes, providing operational definitions and analyzing conditions under which quantum processes appear classical.

## Contribution

It introduces an operational definition of incoherent dynamics for open systems and clarifies the conditions linking classicality, coherence, and Markovianity in quantum measurements.

## Key findings

- Classicality implies incoherent dynamics for non-degenerate observables.
- For invertible Markovian dynamics, classicality and incoherence are equivalent in non-degenerate cases.
- In degenerate cases, classicality does not necessarily imply incoherent dynamics.

## Abstract

We investigate the role of coherence and Markovianity in finding an answer to the question whether the outcomes of a projectively measured quantum stochastic process are compatible with a classical stochastic process. For this purpose we put forward an operationally motivated definition of incoherent dynamics applicable to any open system's dynamics. For non-degenerate observables described by rank-1 projective measurements we show that classicality always implies incoherent dynamics, whereas the converse is only true for invertible Markovian (but not necessarily time-homogeneous) dynamics. For degenerate observables the picture is somewhat reversed as classicality does no longer suffice to imply incoherent dynamics (even in the invertible Markovian case), while an incoherent, invertible Markovian dynamics still implies classicality.