  We propose an interferometric method for statistically discriminating between
nonorthogonal states in high dimensional Hilbert spaces for use in quantum
information processing. The method is illustrated for the case of photon
orbital angular momentum (OAM) states. These states belong to pairs of bases
that are mutually unbiased on a sequence of two-dimensional subspaces of the
full Hilbert space, but the vectors within the same basis are not necessarily
orthogonal to each other. Over multiple trials, this method allows
distinguishing OAM eigenstates from superpositions of multiple such
eigenstates. Variations of the same method are then shown to be capable of
preparing and detecting arbitrary linear combinations of states in Hilbert
space. One further variation allows the construction of chains of states
obeying recurrence relations on the Hilbert space itself, opening a new range
of possibilities for more abstract information-coding algorithms to be carried
out experimentally in a simple manner. Among other applications, we show that
this approach provides a simplified means of switching between pairs of
high-dimensional mutually unbiased OAM bases.
