Size of quantum superpositions as measured with classical detectors

We propose a criterion which defines whether a superposition of two photonic components is macroscopic. It is based on the ability to discriminate these components with a particular class of "classical" detectors, namely, a photon number measurement with a resolution coarse-grained by noise. We show how our criterion can be extended to a measure of the size of macroscopic superpositions by quantifying the amount of noise that can be tolerated and taking the distinctness of two Fock states differing by N photons as a reference. After applying our measure to several well-known examples, we demonstrate that the superpositions which meet our criterion are very sensitive to phase fluctuations. This suggests that quantifying the macroscopicity of a superposition state through the distinguishability of its components with "classical" detectors not only is a natural measure but also explains why it is difficult to observe superpositions at the macroscopic scale.