The Complete Positivity of Classical Polarization Maps

TL;DR

This paper rigorously derives the most general form of classical polarization transformations, represented as ensembles of Jones matrices, from fundamental physical principles and positive map theory.

Contribution

It provides a formal proof that the general classical polarization transformation can be expressed as an ensemble of Jones matrices, clarifying assumptions in prior work.

Findings

Derived the ensemble representation of polarization maps from physical principles.

Connected polarization transformations with positive map matrix theory.

Clarified the theoretical foundation of classical polarization manipulation.

Abstract

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.