Nonclassicality detection from few Fock-state probabilities

We devise a new class of criteria to certify the nonclassicality of photon- and phonon-number statistics. Our criteria extend and strengthen the broadly used Klyshko's criteria, which require knowledge of only a finite set of Fock-state probabilities. This makes the criteria well-suited to experimental implementation in realistic conditions. Moreover, we prove the completeness of our method in some scenarios, showing that, when only two or three Fock-state probabilities are known, it detects all finite distributions incompatible with classical states. In particular, we show that our criteria detect a broad class of noisy Fock states as nonclassical, even when Klyshko's do not. The method is directly applicable to trapped-ion, superconducting circuits, and optical and optomechanical experiments with photon-number resolving detectors. This work represents a significant milestone towards a complete characterisation of the nonclassicality accessible from limited knowledge of the Fock-state probabilities.