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We introduce unitary quantum gates for photon pair creation in spontaneousparametric down-conversion nonlinear crystals (NLs) and for photon pathalignment. These are the two key ingredients for the method of "inducedcoherence without induced emission" and many ensuing variations thereof. Thedifficulty in doing so stems from an apparent mixing of the mode picture (suchas the polarization of photons) and the Fock picture (such as the existence ofthe photons). We illustrate utility of these gates by obtaining quantumcircuits for the experimental setups of the frustrated generation of photonpairs, identification of a point-like object with undetected photons, andcreation of a Bell state. We also introduce an effective nonunitary descriptionfor 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 usedto create superposition of given quantum states in a modular fashion.

Quantum gate description for induced coherence without induced emission and related phenomena