Statistics of projective measurement on a quantum probe as a witness of noncommutativity of algebra of a probed system

Abstract: We consider a quantum probe $P$ undergoing pure dephasing due to its interaction with a quantum system $S$. The dynamics of $P$ is then described by a well-defined sub-algebra of operators of $S,$ i.e. the "accessible" algebra on $S$ from the point of view of $P.$ We consider sequences of $n$ measurements on $P,$ and investigate the relationship between Kolmogorov consistency of probabilities of obtaining sequences of results with various $n,$ and commutativity of the accessible algebra. For a finite-dimensional $S$ we find conditions under which the Kolmogorov consistency of measurement on $P,$ given that the state of $S$ can be arbitrarily prepared, is equivalent to the commutativity of this algebra. These allow us to describe witnesses of nonclassicality (understood here as noncommutativity) of part of $S$ that affects the probe. For $P$ being a qubit, the witness is particularly simple: observation of breaking of Kolmogorov consistency of sequential measurements on a qubit coupled to $S$ means that the accessible algebra of $S$ is noncommutative.