  Multi-photon hyperentangement is of fundamental importance in optical quantum
information processing. Existing theory and experiment producing multi-photon
hyperentangled states have until now relied on the outcome post-selection, a
procedure where only the measurement results corresponding to the desired state
are considered. Such approach severely limits the usefulness of the resulting
hyperentangled states. We present the protocols of direct production of three-
and four-photon hyperentanglement and extend the approach to an arbitrary
number of photons through a straightforward cascade of spontaneous parametric
down-conversion (SPDC) sources. The generated multi-photon hyperentangled
states are encoded in polarization-spatial modes and polarization-time bin
degrees of freedom, respectively. Numerical calculation shows that if the
average photon number $\mu$ is set to 1, the down conversion efficiency is
$7.6*10^{-6}$ and the repetition frequency of the laser is $10^9$ Hz, the
number of the generation of three-photon and four-photon hyperentanglement
after cascading can reach about $5.78*10^{-2}$ and $4.44*10^{-7}$ pairs per
second, respectively. By eliminating the constraints of outcome post-selection,
our protocols may represent important progresses for multi-photon
hyperentangement generation and providing a pivotal role in future multi-party
and high-capacity communication networks.
