Limits on entropic uncertainty relations for 3 and more MUBs

TL;DR

This paper investigates the limits of entropic uncertainty relations for measurements in three or more mutually unbiased bases in prime dimensions, showing that improvements over known two-basis relations are minimal.

Contribution

It demonstrates that for three or more MUBs, the optimal entropic uncertainty bounds are only marginally better than the classical two-basis bounds.

Findings

Optimal bounds are nearly the same as the Maassen-Uffink relation for 2 bases.

Adding more MUBs does not significantly tighten the uncertainty bounds.

The results apply to standard constructions of MUBs in prime dimensions.

Abstract

We consider entropic uncertainty relations for outcomes of the measurements of a quantum state in 3 or more mutually unbiased bases (MUBs), chosen from the standard construction of MUBs in prime dimension. We show that, for any choice of 3 MUBs and at least one choice of a larger number of MUBs, the best possible entropic uncertainty relation can be only marginally better than the one that trivially follows from the relation by Maassen and Uffink (PRL, 1987) for 2 bases.