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Spontaneous parametric down conversion (PDC), in the perturbative limit, canbe considered as a probabilistic splitting of one input photon into two outputphotons. Conversely, sum-frequency generation (SFG) implements the reverseprocess of combining two input photons into one. Here we show that asingle-photon projective measurement in the temporal-mode basis of the outputphoton of a two-photon SFG process effects a generalized measurement on theinput two-photon state. We describe the positive-operator-valued measure (POVM)associated with such a measurement, and show that its elements are proportionalto the two-photon states produced by the time-reversed PDC process. Such adetection acts as a joint measurement on two photons, and is thus an importantcomponent of many quantum information processing protocols relying on photonicentanglement. Using the retrodictive approach, we analyze the properties of thetwo-photon POVM that are relevant for quantum protocols exploiting two-photonstates and measurements.

A positive operator-valued measure for two-photon detection via sum-frequency generation