Simulating quantum measurements without superposition devices

Superposition is the core feature that sets quantum theory apart from classical physics. Here, we investigate whether sets of quantum measurements can be modelled by using only devices that are classical, in the sense that they only resolve orthogonal measurement outcomes. This leads us to introduce classical measurement models, which we show to be intermediate between the notion of commutative measurements and joint measurability. Towards understanding these models we (i) identify exact noise and loss rates at which all projective measurements admit a classical model, (ii) propose numerical methods to construct classical models for finite sets of measurements, and (iii) show how to construct witnesses of genuine superposition properties in quantum measurements. In addition, we show that classical measurement models also have operational implications in non-disturbance tasks where sequential quantum measurements are implemented with classical side-information. Our work provides a new approach to the role of superposition in quantum measurements.