# Tight entropic uncertainty relations for systems with dimension three to five

## TL;DR

This paper derives tight entropic uncertainty relations for systems of dimensions three to five, focusing on spin observables and mutually unbiased bases, with bounds that improve upon previous results.

## Contribution

The paper introduces new tight entropic uncertainty bounds for specific observables in low-dimensional quantum systems, identifying states that attain these bounds.

## Key findings

- Most bounds are stronger than previous ones.
- Explicit forms of states attaining the bounds are provided.
- Applicable to systems with dimensions 3, 4, and 5.

## Abstract

We consider two (natural) families of observables $O_k$ for systems with dimension $d=3,4,5$: the spin observables $S_x$, $S_y$ and $S_z$, and the observables that have mutually unbiased bases as eigenstates. We derive tight entropic uncertainty relations for these families, in the form $\sum_kH(O_k)\geqslant\alpha_d$, where $H(O_k)$ is the Shannon entropy of the measurement outcomes of $O_k$ and $\alpha_d$ is a constant. We show that most of our bounds are stronger than previously known ones. We also give the form of the states that attain these inequalities.