  The problem of unambiguously distinguishing among nonorthogonal but linearly
independent quantum states can be solved by mapping the set of nonorthogonal
quantum states onto a set of orthogonal ones, which can then be distinguished
without error. Such nonunitary transformations can be performed conditionally
on quantum systems; a unitary transformation is carried out on a larger system
of which the system of interest is a subsytem, a measurement is performed, and
if the proper result is obtained, the desired nonunitary transformation will
have been performed on the subsystem. We show how to construct generalized
interferometers (multiports), which when combined with measurements on some of
the output ports, implement nonunitary transformations of this type. The input
states are single-photon states in which the photon is divided among several
modes. A number of explicit examples of distinguishing among three
nonorthogonal states are discussed, and the networks that optimally distinguish
among these states are presented.
