Nearly tight weighted 2-designs in complex projective spaces of every dimension

TL;DR

This work addresses Zauner's conjecture on the existence of $d^2$ equi-distance points (SICs) in $\mathbb{CP}^{d-1}$ by connecting SIC existence to the size of weighted projective $2$-designs through the entanglement-breaking rank $n(d)$. It employs the Bodmann--Haas construction together with dense Sidon sets to produce small weighted $2$-designs in $\mathbb{CP}^{|S|-1}$, yielding upper bounds on $n(d)$. The main result shows $n(d) \le d^2 + O(d^{1.525})$, the first general bound that is $o(d^4)$, derived from bounding the minimal group containing a Sidon set of size $d$ via the least prime $p(d)\ge d$ and the Baker–Harman–Pintz estimate $p(d) \le d+O(d^{0.525})$. The method highlights the role of specific Sidon-set families (e.g., Singer, Bose, Spence, Hughes) and clarifies limitations, pointing to progress via new exact SICs or sharper prime-gap results.

Abstract

We use dense Sidon sets to construct small weighted projective 2-designs. This represents quantitative progress on Zauner's conjecture.

Theorems & Definitions (12)

• Proposition 1
• Definition 2
• Proposition 3
• proof
• Theorem 4
• Definition 5
• Definition 6
• Theorem 7
• proof
• Corollary 8