Zauner's conjecture

Remarks

Zauner posed the question in his 1999 doctoral thesis at the University of Vienna. He also proposed a stronger form. Let $X$ and $Z$ be the shift and clock operators on $\C^d$, with $X e_j = e_{j+1}$ and $Z e_j = \omega^j e_j$, where indices are taken modulo $d$ and $\omega = e^{2\pi i/d}$. The stronger form says that some unit vector $\psi$ has orbit $\set{X^a Z^b \psi : 0 \le a, b < d}$ forming a SIC-POVM, up to phases. It adds that such a $\psi$ can be chosen as an eigenvector of a certain unitary of order $3$ in the Clifford group, now called the Zauner unitary.

The conjecture is open. For $d = 2$ the regular tetrahedron on the Bloch sphere gives a SIC-POVM. Exact solutions are known in many further dimensions, and numerical solutions in many more. The known exact solutions have been linked to ray class fields of real quadratic number fields, a link to Hilbert's twelfth problem.