Tight entropic uncertainty relations for systems with dimension three to five

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.