Unconditional construction of SICs in an infinite sequence of dimensions
This text mentions work from 2024 on. Those parts were partly written without a live literature search and have not been checked by an editor yet.
In plain words
A number-theory recipe produces SICs in many dimensions of a special form, but it relies on unproved conjectures. Proving it works for infinitely many dimensions would be a first step to the full answer.
Precise statement
Prove, without assuming the Stark conjectures, that SICs exist for infinitely many d, for example all prime $d = n^2 + 3$, where Appleby, Bengtsson, Grassl, Harrison and McConnell (J. Math. Phys. 63, 112205, 2022, arXiv:2112.05552) construct SIC fiducials from Stark units of real quadratic fields and obtain solutions in thirteen prime dimensions, the largest $d = 19603$.
What would settle it
An unconditional proof that the construction yields valid SICs in an infinite family of dimensions.
Status in the literature
Unverified note: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
A 2025 extension to dimensions $n^{2}+3=4p,\ p\ \text{prime}$, reached $d=39604$ (Bengtsson, Grassl and McConnell, J. Math. Phys. 66, 082202, 2025), still conditional.
Parent problems
- Special case of Do SIC-POVMs exist in every finite dimension? · open · from revision 1
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