Which process states exist a multi-processor machine?
More states aren't technically needed, depending on an operating system's implementation. 

A kernel can have a task scheduler running on each processor. This allows for the use of the same process states, since each scheduler is running independent of each other. The kernel can have "load balancing" between the processors so that not too many processes are queued-up on one processor

xv6 (UNIX V6 port for x86 architecture) is a bare-bones kernel that does just this, and so, it has the same process states like in a single-processor environment.