The ensemble of prepared systems, in a definite pure state, then consists of a set of individual systems, all having one and the same definite energy, but each having a different quantum mechanical phase, regarded as probabilistically random. The wave function, however, does have a definite phase, and thus specification by a wave function is more detailed than specification by state as prepared.
The ensemble of prepared systems, in their pure state, consists of individual systems with the same energy but different quantum mechanical phases, which are randomly distributed.  However, the wave function possesses a definite phase, making it a more precise description than specifying the system's state.