  We present two complete maximally hyperentangled state analysis protocols for
photons entangled in the polarization and spatial-mode degrees of freedom. The
first protocol is a hyperentangled Bell state analysis scheme for two photons
and the second is a hyperentangled Greenberger-Horne- Zeilinger (GHZ) state
analysis scheme for three photons. In each scheme, a set of mutually orthogonal
hyperentangled basis states are completely and deterministically discriminated
with the aid of cross-Kerr nonlinearities and linear optics. We also generalize
the schemes to unambiguously analyze the N-photon hyperentangled GHZ state.
Compared with previous protocols, our schemes greatly simplify the
discrimination process and reduce the requirements on nonlinearities by using
the measured spatial-mode state to assist in the analysis of the polarization
state. These advantages make our schemes useful for practical applications in
long-distance high capacity quantum communication.
