Certification of stellar ranks of quantum states of light with a pair of click detectors

Abstract

Stellar rank of quantum state of light quantifes the amount of non-Gaussian resources required for its generation. One popular and practical approach to certification of stellar rank is based on measurement of click statistics with an array of binary detectors that can only distinguish the presence and absence of photons. Specifically, it was shown that measurements with an array of m+1 detectors allow one to certify stellar rank m of approximate Fock state $|m\rangle$, even when the state is subjected to losses or certain noise. Here we address the question as to how many click detectors are in principle required to certify stellar ranks higher than one. We show that two click detectors arranged in a Hanbury Brown-Twiss setup suffice. Interestingly, detection of stellar ranks higher than one is greatly facilitated by making the total detection efficiency of the detectors sufficiently low but well calibrated. Losses affect the response of the detection scheme in a way that can be exploited to certify stellar rank of higher Fock states. We explicitly construct the corresponding stellar rank witnesses and discuss dependence of the stellar rank thresholds on parameters of the considered setup. Our results reveal that it is possible to certify stellar ranks higher than 1 even with a minimalistic scheme that provides only very coarse-grained information about the photon number statistics of the characterized state.