  We introduce unitary quantum gates for photon pair creation in spontaneous
parametric down-conversion nonlinear crystals (NLs) and for photon path
alignment. These are the two key ingredients for the method of "induced
coherence without induced emission" and many ensuing variations thereof. The
difficulty in doing so stems from an apparent mixing of the mode picture (such
as the polarization of photons) and the Fock picture (such as the existence of
the photons). We illustrate utility of these gates by obtaining quantum
circuits for the experimental setups of the frustrated generation of photon
pairs, identification of a point-like object with undetected photons, and
creation of a Bell state. We also introduce an effective nonunitary description
for the action of NLs in experiments where all the NLs are pumped coherently.
As an example, by using this simplifying picture, we show how NLs can be used
to create superposition of given quantum states in a modular fashion.
