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

Abstract

The entropic uncertainty relation for sets of $N+1$ complementary observables $\{A_r\}$ existing in $N$-dimensional Hilbert space, $\sum_r H(A_r) \geq (N+1) \ln((N+ 1)/2)$, 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 calculation, this is conjectured to also be the case for odd $N>3$.