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

Stellar rank of quantum states of light quantifies the amount of non-Gaussian resources required for their 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 allows one to certify stellar rank of approximate Fock state |m>, even when the state is subjected to losses or certain noise. Here we address the question how many click detectors are in principle required to certify high stellar ranks. We show that two click detectors arranged in a Hanbury Brown-Twiss setup suffice. Interestingly, detection of higher stellar rank 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 high 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 high stellar ranks even with a minimalistic scheme that provides only very coarse grained information about the photon number statistics of the characterized state.