  Non-commutativity is one of the most elementary non-classical features of
quantum observables. Here we propose a method to detect non-commutativity of
interaction Hamiltonians of two probe objects coupled via a mediator. If these
objects are open to their local environments, our method reveals
non-decomposability of temporal evolution into a sequence of interactions
between each probe and the mediator. The Hamiltonians or Lindblad operators can
remain unknown throughout the assessment, we only require knowledge of the
dimension of the mediator. Furthermore, no operations on the mediator are
necessary. Technically, under the assumption of decomposable evolution, we
derive upper bounds on correlations between the probes and then demonstrate
that these bounds can be violated with correlation dynamics generated by
non-commuting Hamiltonians, e.g., Jaynes-Cummings coupling. An intuitive
explanation is provided in terms of multiple exchanges of a virtual particle
which lead to the excessive accumulation of correlations. A plethora of
correlation quantifiers are helpful in our method, e.g., quantum entanglement,
discord, mutual information, and even classical correlation. Finally, we
discuss exemplary applications of the method in quantum information: the
distribution of correlations and witnessing dimension of an object.
