Existence of SIC POVMs in an infinite sequence of dimensions

Construct a family of symmetric informationally complete positive operator-valued measures (SIC POVMs) for an infinite sequence of Hilbert space dimensions N1, N2, N3, …, by finding for each dimension N an ordered set of N^2 unit vectors in C^N whose pairwise overlaps satisfy |⟨ψj|ψk⟩|^2 = (N δjk + 1)/(N + 1).

Background

A symmetric informationally complete positive operator-valued measure (SIC POVM) in dimension N is specified by N2 pure states in CN with equal pairwise overlaps that define an informationally optimal measurement and a simplex embedded in the set of density matrices. Zauner conjectured that for every dimension N there exists a fiducial vector generating a SIC through the Weyl–Heisenberg group.

Numerical and analytic solutions are known in many specific dimensions, but a general proof of existence in all dimensions remains elusive. Constructing SIC POVMs in any infinite sequence of dimensions would represent decisive progress toward Zauner’s conjecture and has deep links to algebraic number theory.

References

However, in spite of a considerable research effort, the general conjecture of Zauner remains unproven.

— Five open problems in quantum information  (2002.03233 - Horodecki et al., 2020) in Section: Discrete structures in the Hilbert space; Subsection: Existence of SIC POVMs