Positive-operator-valued-measure view of the ensemble approach to polarization optics

Abstract

The statistical ensemble formalism of Kim et al [J. Opt. Soc. Am. A4, 433 (1987)] offers a realistic model for characterizing the effect of stochastic nonimage-forming optical media on the state of polarization of transmitted light. With suitable choice of the Jones ensemble, various Mueller transformations-some of which are hitherto unknown-are deduced. It is observed that the ensemble approach is formally identical to the positive-operator-valued measures (POVMs) on the quantum density matrix. This observation, in combination with the recent suggestion by Ahnert and Payne [Phys. Rev. A71, 012330-1 (2005)]-in the context of generalized quantum measurement on single photon polarization states-that linear optics elements can be employed in setting up all possible POVMs enables us to propose a way of realizing different types of Mueller devices.

Quantum Polarimetry

“…Can we circumvent these assumptions using quantum theory? As mentioned before, these questions have been touched on previously by Ahnert and Payne (2005); Aiello et al (2007); Gamel and James (2011); Simon et al (2010);and Sudha et al (2008), but we believe them to remain unanswered.…”

“…It was first noted that the transformations responsible for transforming coherency matrices look like completely positive quantum channels, where the Jones matrices of Eq. ( 27) can be thought of as Kraus operators (Ahnert and Payne, 2005;Aiello et al, 2007;Gamel and James, 2011;Sudha et al, 2008). This hints at the well-known connection with the quantum theory of polarization, viz., that single photons have density matrices described by their coherency matrices 𝚿; the transformations of classical polarization states are akin to transformations of single-photon polarization states.…”

“…with negative "weights" 𝜆 𝑖 . The standard assumption is that only transformations with positive weights 𝜆 𝑖 > 0 are physically viable (Cloude, 1986), which is equivalent to requiring the transformations of the coherency matrix to be completely positive in the sense of Choi's theorem (Ahnert and Payne, 2005;Aiello et al, 2007;Gamel and James, 2011;Sudha et al, 2008). This allows us to interpret nondeterministic transformations as probabilistic mixtures of Jones matrix transformations.…”

“…A flurry of attention revisited this problem motivated by quantum theory (Ahnert and Payne, 2005;Aiello et al, 2007;Gamel and James, 2011;Simon et al, 2010;Sudha et al, 2008), which we have conjectured to truly underlie the necessity of positive weights in Eqs. ( 28) and (31) (Goldberg, 2020).…”

A complete characterization of pre-Mueller and Mueller matrices in polarization optics

“…This corollary is physically important, and has attracted considerable attention [25,26,27,28,29,30,31,32]. In traditional polarization optics, which is formulated for plane waves and not for beams, the state space Ω (pol) is the collection of all Stokes vectors:…”