Unified trajectory criterion for quantum and classical non-Markovianity

We propose a simple criterion for non-Markovianity: a quantum master equation is non-Markovian if and only if its $\mathit{\text{trajectory}}\mathit{\text{set}}$ contains a $\mathit{\text{self-intersecting}}\mathit{\text{trajectory}}$ (defined in the main text). Since self-intersection is invariant under time reversal, our criterion implies that Markovianity itself is time-reversal invariant: This property is not possessed by many existing criteria based on information flow or complete positivity. For a given quantum master equation, the set of trajectories generated from all initial states encodes its essential dynamical features. Thus, Markovianity can be determined directly from the trajectory set, reducing the problem to a general mathematical one: determing the non-Markovianity of the set itself, regardless of whether it originates from a quantum or classical process. Here, we solve this problem and classify the trajectory sets into three types: $\mathit{\text{strictly}}\mathit{\text{Markovian}}$, $\mathit{\text{initial-state}}\mathit{\text{Markovian}}$, and $\mathit{\text{non-Markovian}}$. Through multiple examples, we compare our criterion with existing ones and show how they fail to capture Markovianity and non-Markovianity in various cases.