Unified treatment of the total angular momentum of single photons via generalized quantum observables

In this paper, we provide a consistent framework to address the notorious difficult decomposition of the single-photon total angular momentum (TAM) into a spin (SAM) and an orbital (OAM) component. We discuss the canonical decomposition into SAM and OAM components, which are the generators of internal and spatial rotations in the space of physical states. We find that those operators are mutually compatible but unsharp quantum observables, therefore Positive Operator-Valued Measures describe their joint measurements. We present another decomposition of the TAM, which we denote as a non-canonical one. The operators resulting from this decomposition are mutually incompatible but sharp quantum observables, thus Projector-Valued Measurements. This fact reflects their consistency with the transversality condition of single-photon wavefunctions, thus explains the underlying physics from a quantum information theoretic view. Furthermore, we discuss the implementations on joint measurements for both decompositions and provide an explicit calculation of all these quantities for circularly polarized Gaussian single-photon states. The difference between the canonical and non-canonical momenta leads to observable differences in higher-order statistical moments.