  Quantum teleportation provides a way to transmit unknown quantum states from
one location to another. In the quantum world, multilevel systems which enable
high-dimensional systems are more prevalent. Therefore, to completely rebuild
the quantum states of a single particle remotely, one needs to teleport
multilevel (high-dimensional) states. Here, we demonstrate the teleportation of
high-dimensional states in a three-dimensional six-photon system. We exploit
the spatial mode of a single photon as the high-dimensional system, use two
auxiliary entangled photons to realize a deterministic three-dimensional Bell
state measurement. The fidelity of teleportation process matrix is
F=0.596\pm0.037. Through this process matrix, we can prove that our
teleportation is both nonclassical and genuine three dimensional. Our work
paves the way to rebuild complex quantum systems remotely and to construct
complex quantum networks.
