Optimal Frames for Polarisation State Reconstruction

TL;DR

This paper analytically determines optimal measurement frames for polarisation state reconstruction, minimizing the condition number and linking optimal analysis states to spherical 2-designs, with implications for nonlinear and quantum measurements.

Contribution

It provides an analytical solution for the minimal condition number in polarisation measurements and connects optimal analysis states to spherical 2-designs, generalizing to higher-order processes.

Findings

Minimum condition number is independent of the number of analysis states, except for m=5.

Optimal states are distributed as spherical 2-designs, including Platonic solids.

Higher order measurements correspond to spherical 2t-designs, extending the concept of mutually unbiased bases.

Abstract

Complete determination of the polarisation state of light requires at least four distinct projective measurements of the associated Stokes vector. Stability of state reconstruction, however, hinges on the condition number $\kappa$ of the corresponding instrument matrix. Optimisation of redundant measurement frames with an arbitrary number of analysis states, $m$, is considered in this Letter in the sense of minimisation of $\kappa$. The minimum achievable $\kappa$ is analytically found and shown to be independent of $m$, except for $m=5$ where this minimum is unachievable. Distribution of the optimal analysis states over the Poincar\'e sphere is found to be described by spherical 2-designs, including the Platonic solids as special cases. Higher order polarisation properties also play a key role in nonlinear, stochastic and quantum processes. Optimal measurement schemes for nonlinear measurands of degree $t$ are hence also considered and found to correspond to spherical $2t$-designs, thereby constituting a generalisation of the concept of mutually unbiased bases.