  We analyse the problem of finding sets of quantum states that can be
deterministically discriminated. From a geometric point of view this problem is
equivalent to that of embedding a simplex of points whose distances are maximal
with respect to the Bures distance (or trace distance). We derive upper and
lower bounds for the trace distance and for the fidelity between two quantum
states, which imply bounds for the Bures distance between the unitary orbits of
both states. We thus show that when analysing minimal and maximal distances
between states of fixed spectra it is sufficient to consider diagonal states
only. Hence considering optimal discrimination, given freedom up to unitary
orbits, it is sufficient to consider diagonal states. This is illustrated
geometrically in terms of Weyl chambers.
