Tight entropic uncertainty relations for systems with dimension three to five
Alberto Riccardi, Chiara Macchiavello, Lorenzo Maccone · Physical Review A · 2017
We consider two (natural) families of observables ${O}_{k}$ for systems with dimension $d=3,4,5$: the spin observables ${S}_{x},\phantom{\rule{0.16em}{0ex}}{S}_{y}$, 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 ${\ensuremath{\sum}}_{k}H({O}_{k})\ensuremath{\ge}{\ensuremath{\alpha}}_{d}$, where $H({O}_{k})$ is the Shannon entropy of the measurement outcomes of ${O}_{k}$ and ${\ensuremath{\alpha}}_{d}$ 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.