Pure states of maximum uncertainty with respect to a given POVM

TL;DR

This paper investigates the pure quantum states that maximize measurement outcome uncertainty for symmetric POVMs and SIC-POVMs, revealing states that produce the highest randomness in measurement results.

Contribution

It identifies the pure states that maximize measurement uncertainty for all symmetric POVMs in dimension 2 and all SIC-POVMs in any dimension.

Findings

Maximizers of measurement uncertainty for symmetric POVMs in dimension 2.

Maximizers for SIC-POVMs in arbitrary dimensions.

Provides a characterization of states with maximum measurement entropy.

Abstract

One of the differences between classical and quantum world is that in the former we can always perform a measurement that gives certain outcomes for all pure states, while such a situation is not possible in the latter. The degree of randomness of the distribution of the measurement outcomes can be quantified by the Shannon entropy. While it is well known that this entropy, as a function of quantum states, needs to be minimized by some pure states, we would like to address the question how 'badly' can we end by choosing initially any pure state, i.e., which pure states produce the maximal amount of uncertainty under given measurement. We find these maximizers for all highly symmetric POVMs in dimension 2, and for all SIC-POVMs in any dimension.