  Birefringent materials or nanostructures that introduce phase differences
between two linear polarizations underpin the operation of wave plates for
polarization control of light. Here we develop metasurfaces realizing a
distinct class of complex-birefringent wave plates, which combine polarization
transformation with a judiciously tailored polarization-dependent phase
retardance and amplitude filtering via diffraction. We prove that the presence
of loss enables the mapping from any chosen generally non-orthogonal pair of
polarizations to any other pair at the output. We establish an optimal
theoretical design-framework based on pairwise nanoresonator structures and
experimentally demonstrate unique properties of metasurfaces in the
amplification of small polarization differences and polarization coupling with
unconventional phase control. Furthermore, we reveal that these metasurfaces
can perform arbitrary transformations of biphoton polarization-encoded quantum
states, including the modification of the degree of entanglement. Thereby, such
flat devices can facilitate novel types of multi-functional polarization optics
for classical and quantum applications.
