  We show that the states generated by a three-mode spontaneous parametric
downconversion (SPDC) interaction Hamiltonian possess tripartite entanglement
of a different nature to other paradigmatic three-mode entangled states
generated by the combination of two-mode SPDCs interactions. While two-mode
SPDC generates gaussian states whose entanglement can be characterized by
standard criteria based on two-mode quantum correlations, these criteria fail
to capture the entanglement generated by three-mode SPDC. We use criteria built
from three-mode correlation functions to show that the class of states recently
generated in a superconducting-circuit implementation of three-mode SPDC
ideally have tripartite entanglement, contrary to recent claims in the
literature. These criteria are suitable for triple SPDC but we show that they
fail to detect tripartite entanglement in other states which are known to
possess it, which illustrates the existence of two fundamentally different
notions of tripartite entanglement in three-mode continuous variable systems.
