  We lay down a general scheme to quantify the amount of genuine tripartite
entanglement present in the spatial and energy-time degrees of freedom of
entangled photon triplets using a resource-based measure known as the
tripartite entanglement of formation. Quantifying genuine tripartite
entanglement relative to a number of maximally entangled three-qubit
Greenberger-Horne-Zeilinger (GHZ) states called gebits, the tripartite
entanglement of formation serves as a basis of comparison between different
tripartite entangled states of differing dimension. Demonstrating genuine
tripartite entanglement is doubly challenging because it is not enough to show
that each party is inseparable from the other two (which is sufficient only for
pure states). Instead, one must rule out all mixtures of all combinations of
biseparable states from describing the tripartite state. To meet this
challenge, we use entropic measures of the statistics of tripartite systems to
simultaneously bound the correlations each party has with the other two, and in
so doing place a lower limit to the tripartite entanglement of formation. Even
though our measure never over-estimates the entanglement present, we estimate
the effectiveness of our technique by determining the exact tripartite
entanglement of a triple-gaussian triphoton wavefunction with the same
correlations as seen in photon triplets generated in third-order spontaneous
parametric down-conversion (SPDC). Between these two efforts, we show that a
substantial amount of tripartite entanglement exists in both the spatial and
energy-time degrees of freedom of these systems, and propose an experiment that
can measure them.
