Quantitative Nonclassicality of Mediated Interactions

In a plethora of physical situations, one can distinguish a mediator—a system that couples other, noninteracting, systems. Often, the mediator itself is not directly accessible to experimentation, yet it is interesting and sometimes crucial to understand if it admits nonclassical properties. An example of this sort that has recently been enjoying considerable attention is that of two quantum masses coupled via a gravitational field. It has been argued that the gain of quantum entanglement between the masses indicates nonclassicality of the states of the whole tripartite system. Here, we focus on the nonclassical properties of the involved interactions rather than the states. We derive inequalities the violation of which indicates noncommutativity and nondecomposability (open-system generalization of noncommuting unitaries) of interactions through the mediators. The derivations are based on properties of general quantum formalism and make minimalistic assumptions about the studied systems; in particular, the interactions can remain uncharacterized throughout the assessment. Furthermore, we also present conditions that solely use correlations between the coupled systems, excluding the need to measure the mediator. Next, we show that the amount of violation places a lower bound on suitably defined degree of nondecomposability. This makes the methods quantitative and at the same time experiment ready. We give applications of these techniques in two different fields: for detecting the nonclassicality of gravitational interaction and in bounding the Trotter error in quantum simulations.