We introduce unitary quantum gates for photon pair creation in spontaneous parametric downconversion
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 pointlike 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.
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