  We present a theoretical study of minimum error probability discrimination,
using quantum- optical probe states, of M optical phase shifts situated
symmetrically on the unit circle. We assume ideal lossless conditions and full
freedom for implementing quantum measurements and for probe state selection,
subject only to a constraint on the average energy, i.e., photon number. In
particular, the probe state is allowed to have any number of signal and
ancillary modes, and to be pure or mixed. Our results are based on a simple
criterion that partitions the set of pure probe states into equivalence classes
with the same error probability performance. Under an energy constraint, we
find the explicit form of the state that minimizes the error probability. This
state is an unentangled but nonclassical single-mode state. The error
performance of the optimal state is compared with several standard states in
quantum optics. We also show that discrimination with zero error is possible
only beyond a threshold energy of (M - 1)/2. For the M = 2 case, we show that
the optimum performance is readily demonstrable with current technology. While
transmission loss and detector inefficiencies lead to a nonzero erasure
probability, the error rate conditional on no erasure is shown to remain the
same as the optimal lossless error rate.
