Graph states of prime-power dimension from generalized CNOT quantum circuit

We construct multipartite graph states whose dimension is the power of a prime number. This is realized by the finite field, as well as the generalized controlled-NOT quantum circuit acting on two qudits. We propose the standard form of graph states up to local unitary transformations and particle permutations. The form greatly simplifies the classification of graph states as we illustrate up to five qudits. We also show that some graph states are multipartite maximally entangled states in the sense that any bipartition of the system produces a bipartite maximally entangled state. We further prove that 4-partite maximally entangled states exist when the dimension is an odd number at least three or a multiple of four.