Modelling quantum measurements without superposition

Abstract: 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 operationally classical, in the sense that they have no superposition properties. This leads us to propose classical measurement models, which we show to be stronger than commutative measurements but weaker than joint measurability. We determine both the exact depolarisation noise rate and the measurement loss rate at which the all projective measurements in $d$-dimensional quantum theory admit a classical model. For finite sets of quantum measurements we develop methods both for constructing classical models and for falsifying the existence of such model via prepare-and-measure setups. Furthermore, we show that this concept also has operational implications. For that, we consider whether quantum measurements with classical side-information can be implemented in sequence without causing a disturbance and we show that classical models imply an affirmative answer. Our work sheds light on superposition as a resource for quantum measurement devices.