  A major roadblock for large-scale photonic quantum technologies is the lack
of practical reliable certification tools. We introduce an experimentally
friendly - yet mathematically rigorous - certification test for experimental
preparations of arbitrary m-mode pure Gaussian states, pure non-Gaussian states
generated by linear-optical circuits with n-boson Fock-basis states as inputs,
and states of these two classes subsequently post-selected with local
measurements on ancillary modes. The protocol is efficient in m and the inverse
post-selection success probability for all Gaussian states and all mentioned
non-Gaussian states with constant n. We follow the mindset of an untrusted
prover, who prepares the state, and a skeptic certifier, with classical
computing and single-mode homodyne-detection capabilities only. No assumptions
are made on the type of noise or capabilities of the prover. Our technique
exploits an extremality-based fidelity bound whose estimation relies on
non-Gaussian state nullifiers, which we introduce on the way as a byproduct
result. The certification of many-mode photonic networks, as those used for
photonic quantum simulations, boson samplers, and quantum metrology, is now
within reach.
