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.
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