  Spontaneous parametric down conversion (PDC), in the perturbative limit, can
be considered as a probabilistic splitting of one input photon into two output
photons. Conversely, sum-frequency generation (SFG) implements the reverse
process of combining two input photons into one. Here we show that a
single-photon projective measurement in the temporal-mode basis of the output
photon of a two-photon SFG process effects a generalized measurement on the
input two-photon state. We describe the positive-operator-valued measure (POVM)
associated with such a measurement, and show that its elements are proportional
to the two-photon states produced by the time-reversed PDC process. Such a
detection acts as a joint measurement on two photons, and is thus an important
component of many quantum information processing protocols relying on photonic
entanglement. Using the retrodictive approach, we analyze the properties of the
two-photon POVM that are relevant for quantum protocols exploiting two-photon
states and measurements.
