The Complete Positivity of Classical Polarization Maps

Mueller and Jones matrices have been thoroughly studied as mathematical tools to describe the manipulation of the polarization state of classical light. In particular, the most general physical transformation on the polarization state has been represented as an ensemble of Jones matrices, as $\sum_i V_i \Phi V^{\dagger}_i$. But this has generally been directly assumed without proof by most authors. In this Letter, we derive this expression from simple physical principles and the matrix theory of positive maps.