Sharp Total-Support Uncertainty for the Standard Complete Set of Mutually Unbiased Bases

Abstract

We determine the sharp total-support uncertainty for the standard complete set of mutually unbiased bases in every prime dimension at least five, proving that the minimum is \(T_s(p)=p^2-p+2\). This resolves all prime dimensions left open by Fiorentino and Weigert, beginning with dimension eleven, and gives a uniform analytic determination of the optimal lower bound in dimension seven, where the earlier determination was partly numerical. The main new ingredient is a sharp zero-count theorem for arbitrary complex Fourier data supported on the finite parabola. A cyclotomic reduction and truncated local expansion produce a first nonzero finite-field jet that is affine in the transverse variable, reducing the planar zero count to a one-dimensional multiplicity problem. We then classify all extremizing projective states and derive an exact enumeration formula. In dimension seven, this gives analytic four-point extremizers and exactly \(12348\) such rays, completing the analytic description and count left open in the earlier work.