Global optimality of relative rotations for qubit t-design discrimination

Determine whether the relative rotations displayed for the qubit 3-design, 4-design, and 5-design pairs maximize the Chernoff exponent over all relative rotations, and establish the corresponding globally optimal exponents.

Background

The paper constructs pairs of finite uniform qubit t-designs by applying specified rotations to explicit point sets for t=1,2,3,4,5. For each pair, the Chernoff exponent is computed through a maximum-weight perfect-matching problem, yielding the exponents shown in Fig. 1(b).

For t=1 and t=2, the constructed pairs attain the universal upper bound. For t=3,4,5, however, the rotations were selected numerically and the paper reports only the exponents achieved by those particular choices. It therefore leaves unresolved whether alternative relative rotations could produce larger Chernoff exponents, and consequently whether the plotted values are globally optimal within those design families.