Highly nonclassical states of light—such as the approximate Gottesman-Kitaev-Preskill states, states exhibiting cubic nonlinear squeezing, or catlike states—can be generated from experimentally accessible Gaussian states via photon-counting measurements on selected modes, conditioned on specific outcomes of these heralding events. A simple yet important example of this approach involves performing photon number measurements on one mode of a two-mode entangled Gaussian state. The heralding probability of this scheme is a key figure of merit, as it determines the generation rate of the target nonclassical state. In this work, we show that the maximum heralding probability for the two-mode setting can be calculated analytically, and we investigate its dependence on the number of detected photons $n$. Our results show that the number of required experimental trials scales only polynomially with $n$. Generation of highly complex optical quantum states with high stellar rank is thus possible in principle in this setting, given access to sufficiently strong squeezing.

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• Quantum optics
• Quantum state engineering