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Non-commutativity is one of the most elementary non-classical features ofquantum observables. Here we propose a method to detect non-commutativity ofinteraction Hamiltonians of two probe objects coupled via a mediator. If theseobjects are open to their local environments, our method revealsnon-decomposability of temporal evolution into a sequence of interactionsbetween each probe and the mediator. The Hamiltonians or Lindblad operators canremain unknown throughout the assessment, we only require knowledge of thedimension of the mediator. Furthermore, no operations on the mediator arenecessary. Technically, under the assumption of decomposable evolution, wederive upper bounds on correlations between the probes and then demonstratethat these bounds can be violated with correlation dynamics generated bynon-commuting Hamiltonians, e.g., Jaynes-Cummings coupling. An intuitiveexplanation is provided in terms of multiple exchanges of a virtual particlewhich lead to the excessive accumulation of correlations. A plethora ofcorrelation quantifiers are helpful in our method, e.g., quantum entanglement,discord, mutual information, and even classical correlation. Finally, wediscuss exemplary applications of the method in quantum information: thedistribution of correlations and witnessing dimension of an object.

Detecting non-decomposability of time evolution via extreme gain of correlations