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 Ts(p)=p2−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.
Related
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BibTeX
@misc{li2026sharp,
title = {{Sharp Total-Support Uncertainty for the Standard Complete Set of Mutually Unbiased Bases}},
author = {Dongwei Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.16030},
eprint = {2609.16030},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16030},
}
RIS
TY - UNPB
TI - Sharp Total-Support Uncertainty for the Standard Complete Set of Mutually Unbiased Bases
AU - Dongwei Li
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2609.16030
UR - https://arxiv.org/abs/2609.16030
AB - 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.
ER -