STATES AND MINIMAL JOINT UNCERTAINTY FOR COMPLEMENTARY OBSERVABLES IN3-DIMENSIONAL HILBERT-SPACE

The entropic uncertainty relation for sets of N + 1 complementary obse rvables {A(r)} existing in N-dimensional Hilbert space, SIGMA(r) H(A(r )) greater-than-or-equal-to (N + 1)ln((N + 1)/2), is shown to be optim al 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 conjec tured to also be the case for odd N > 3.