States of minimal joint uncertainty for complementary observables in three-dimensional Hilbert space

The entropic uncertainty relation for sets of N+1 complementary observables (A r ) existing in N-dimensional Hilbert space, Sigma r H(A r )>or=(N+1)ln((N+1)/2), is shown to be optimal in the case N=3 by explicit construction of the states for which equality holds. We prove that the lower bound cannot be attained when N is even, and, on the basis of numerical calculations, this is conjectured to also be the case for odd N>3.