  Generalized quantum measurements (also known as POVMs) are of great
importance in quantum information and quantum foundations, but often difficult
to perform. We present an experimental approach which can in principle be used
to perform arbitrary POVMs in a linear-optical context. One of the most
interesting POVMs, the SIC-POVM, is the most compact, set of measurements that
can be used to fully describe a quantum state. We use our technique to carry
out the first experimental characterization of the state of a qutrit using
SIC-POVMs. Because of the highly symmetric nature of this measurement, such a
representation has the unique property that it permits all other measurement
outcomes to be predicted by a simple extension of the classical Bayesian sum
rule, making no use of complex amplitudes or Hilbert-space operators. We
demonstrate this approach on several qutrit states encoded in single photons.
