  We introduce a taxonomy for partially coherent optical fields spanning
multiple degrees of freedom (DoFs) based on the rank of the associated
coherence matrix (the number of non-zero eigenvalues). When DoFs comprise two
spatial modes and polarization, a fourfold classification emerges, with rank-1
fields corresponding to fully coherent fields. We demonstrate theoretically and
confirm experimentally that these classes have heretofore unrecognized
different properties. Specifically, whereas rank-2 fields can always be
rendered separable with respect to its DoFs via a unitary transformation,
rank-3 fields are always non-separable. Consequently, the entropy for a rank-2
field can always be concentrated into a single DoF (thus ridding the other DoF
of statistical fluctuations), whereas some entropy is always 'locked' in one
DoF of a rank-3 field.
