We present a set of criteria to decide whether a given projection measurement can be, in principle, exactly implemented by quantum optical means when only quadratic interaction Hamiltonians are available. These criteria hence apply to any implementation that is based on a linear Bogoliubov transformation of the optical modes, i.e., an arbitrary combination of linear optical elements (multi-port interferometers), multi-mode squeezers, and phase-space displacements. The derivation can be adapted to different detection methods, including photon counting and homodyne detection. Our criteria enable one to obtain easily No-Go theorems for the exact distinguishability of orthogonal quantum states using linear Bogoliubov transformations, auxiliary photons, and conditional dynamics.
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