We demonstrate that the key to realizing arbitrary control over pairs of polarization states of light, i.e. transforming an arbitrarily polarized pair of input states to an arbitrarily polarized pair of output states, is the ability to generate pairs of states with orthogonal polarizations from nonorthogonal pairs of initial states. We then develop a new class of non-Hermitian meta-materials, termed complex birefringent meta-materials, which are able to do exactly this. Such materials could facilitate the detection of small polarization changes in scattering experiments, as well as enable new polarization multiplexing schemes in communications networks.Polarization is one of the fundamental properties of light, and control over the polarization is paramount in many optical communications and imaging applications. In general, the effect of propagation through any media on the polarization of an incident electromagnetic signal can be described as |β = S(z)|α , where |α and |β are the input and output polarization states, respectively, and S(z) is a 2 × 2 matrix that depends on the properties of the medium, as well as the propagation distance z. Conventionally, the polarization of a signal is manipulated through the use of birefringent materials [1][2][3][4][5][6][7]. For lossless birefringent media, with proper choice of material parameters and propagation distance, it is always possible to convert an input polarization |α 1 to an arbitrary output polarization |β 1 . However, once the response to |α 1 is determined, the output polarization |β 2 = S|α 2 is no longer arbitrary for any other input polarization |α 2 . This is because S is unitary in lossless media, and thus β 2 |β 1 = α 2 |α 1 .In this Letter, we seek to overcome the limitation of conventional birefringent media by developing a class of meta-materials which enable arbitrary control over pairs of polarization states. By arbitrary control, we demand that for a pair of arbitrary input polarizations |α 1 and |α 2 , one can generate an arbitrary pair of output polarizations |β 1 and |β 2 . Achieving such polarization control has significant implications for a wide range of technologies. For example, with this capability one can map two polarizations that are close to each other into two orthogonal polarizations, which may facilitate the detection of small polarization changes, such as those arising from the imaging of biological tissues [8,9] and thin films [10]. Likewise, the ability to completely separate nonorthogonal polarization states could enable new multiplexing schemes in optical communications networks beyond what is currently possible [11,12].We first show that the key step for achieving arbitrary control over pairs of polarization states is to develop a class of meta-materials which are capable of performing the following polarization transformation as denoted byHere we assume propagation along the z-axis, and label the polarization states in terms of the electric field components in the xy-plane as |E x , E y . | ± θ denote the two polarizatio...