Tight entropic uncertainty relations for systems with dimension three to five

We consider two (natural) families of observables Ok for systems with dimension d=3,4,5: the spin observables Sx,Sy, and Sz, and the observables that have mutually unbiased bases as eigenstates. We derive tight entropic uncertainty relations for these families, in the form ∑kH(Ok)≥αd, where H(Ok) is the Shannon entropy of the measurement outcomes of Ok and αd 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.