  For partially coherent optical fields in which a single binary degree of
freedom (DoF) is relevant, such as polarization, entropy uniquely identifies
the class of optical fields that can be converted into each other via unitary
transformations. However, when multiple DoFs are taken into consideration,
entropy no longer serves this purpose. We investigate the structure of the
family of iso-entropy partially coherent optical fields defined by two binary
DoFs (polarization and two spatial modes) and described by a 4x4 coherence
matrix G. We find that the rank of G (the number of its non-zero eigenvalues)
plays a critical role in this context: whereby any pair of iso-entropy rank-2
fields can be converted into each other unitarily, this is not necessarily the
case for a pair of rank-3 or rank-4 fields. Furthermore, unitary
transformations between iso-entropy fields of different ranks are strictly
forbidden. Instead, such conversions require entropy-maintaining non-unitary
transformations that potentially combine filtering projections and randomizing
operations. We experimentally synthesize partially coherent iso-entropy optical
fields of all ranks, and tomographically reconstruct their coherence matrices.
Moreover, we steer the coherence matrix over iso-entropy trajectories that
maintain a fixed rank (intra-rank conversion) or that involve changes in the
rank (inter-rank conversion). These findings offer a new perspective for the
potential utility of partially coherent light in optical communications and
sensing.
