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

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.