Optimal weak value measurements: Pure states

TL;DR

This paper investigates the optimality of weak value measurements for pure state tomography, demonstrating that mutually unbiased bases yield optimal measurements across finite-dimensional Hilbert spaces and exploring the geometric structure of weak values.

Contribution

It extends the concept of measurement optimality to weak value tomography, proving mutual unbiasedness as optimal and analyzing the geometric properties of weak values in arbitrary dimensions.

Findings

Mutually unbiased bases are optimal for weak value measurements.

Derived the Ka"ehler potential for N-dimensional state spaces.

Provided geometric insights into weak value coordinates.

Abstract

We apply the notion of \emph{optimality} of measurements for state determination(tomography) as originally given by Wootters and Fields to \emph{weak value tomography} of \emph{pure states}. They defined measurements to be optimal if they 'minimised' the effects of statistical errors. For technical reasons they actually maximised the state averaged information, precisely quantified as the negative logarithm of 'error volume'. In this paper we optimise both the state averaged information as well as error volumes. We prove, for Hilbert spaces of arbitrary (finite) dimensionality, that varieties of weak value measurements are optimal when the post-selected bases are \emph{mutually unbiased} with respect to the eigenvectors of the observable being measured. We prove a number of important results about the geometry of state spaces when expressed through the weak values as coordinates. We derive an expression for the Ka\"ehler potential for the N-dimensional case with the help of which we give an exact treatment of the arbitrary-spin case.