Do SIC-POVMs exist in every finite dimension?
In plain words
A SIC is a set of $d^2$ quantum states in dimension $d$ that are all equally far from one another. They have been found in many dimensions, but nobody has proven they exist in all.
Precise statement
For every integer $d \ge 2$, do there exist $d^{2}$ unit vectors $\psi_j$ in $C^d$ with $\mid\langle\psi_j\mid\psi_k\rangle\mid^{2} = 1/(d+1)$ for all $j != k$? Zauner's conjecture further asserts a Weyl-Heisenberg covariant solution whose fiducial vector is fixed by an order-3 Clifford unitary. Exact solutions are known in all $d$ up to about 50 and in some much larger $d$, numerical ones in all d up to about 150 (figures approximate).
What would settle it
A proof of existence for all d, or a dimension with no SIC.
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 route via the Stark conjectures of algebraic number theory makes existence conditional for some infinite families (Appleby, Flammia, McConnell and Yard 2017 onward); no unconditional proof as of 2026.
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