  Two-photon Hong-Ou-Mandel (HOM) interference is a fundamental quantum effect
with no classical counterpart. The exiting researches on two-photon
interference were mainly limited in one degree of freedom (DoF), hence it is
still a challenge to realize the quantum interference in multiple DoFs. Here we
demonstrate the HOM interference between two hyper-entangled photons in two
DoFs of polarization and orbital angular momentum (OAM) for all the sixteen
hyper-entangled Bell states. We observe hyper-entangled two-photon interference
with bunching effect for ten symmetric states (nine Boson-Boson states, one
Fermion-Fermion state) and anti-bunching effect for six anti-symmetric states
(three Boson-Fermion states, three Fermion-Boson states). More interestingly,
expanding the Hilbert space by introducing an extra DoF for two photons enables
to transfer the unmeasurable external phase in the initial DoF to a measurable
internal phase in the expanded two DoFs. We directly measured the symmetric
exchange phases being $0.012 \pm 0.002$, $0.025 \pm 0.002$ and $0.027 \pm
0.002$ in radian for the three Boson states in OAM and the anti-symmetric
exchange phase being $0.991 \pi \pm 0.002$ in radian for the other Fermion
state, as theoretical predictions. Our work may not only pave the way for more
wide applications of quantum interference, but also develop new technologies by
expanding Hilbert space in more DoFs.
