  Multipartite high-dimensional entanglement presents different physics from
multipartite two-dimensional entanglement. However, how to prepare multipartite
high-dimensional entanglement is still a challenge with linear optics. In this
paper, a multiphoton GHZ state with arbitrary dimensions preparation protocol
is proposed in optical systems. In this protocol, we use auxiliary
entanglements to realize a high-dimensional entanglement gate, so that
high-dimensional entangled pairs can be connected into a multipartite
high-dimensional GHZ state. Specifically, we give an example of using photons'
path degree of freedom to prepare a 4-particle 3-dimensional GHZ state. Our
method can be extended to other degrees of freedom and can generate arbitrary
GHZ entanglement in any dimension.
