SIC-POVMs and the Stark conjectures

TL;DR

This paper links the existence of SIC-POVMs in certain dimensions to Stark conjectures over real quadratic fields, providing explicit formulas and numerical verification for dimensions 5, 11, 17, and 23.

Contribution

It introduces a novel construction of SIC-POVMs based on algebraic units in ray class fields and connects this to Stark conjectures, offering explicit formulas and exact solutions in specific dimensions.

Findings

Constructs SIC-POVMs in dimensions 5, 11, 17, and 23.

Provides an explicit analytic formula for SIC-POVM construction.

First exact solution to SIC-POVM problem in dimension 23.

Abstract

The existence of a set of d^2 pairwise equiangular complex lines (equivalently, a SIC-POVM) in d-dimensional Hilbert space is currently known only for a finite set of dimensions d. We prove that, if there exists a set of real units in a certain ray class field (depending on d) satisfying certain congruence conditions and algebraic properties, a SIC-POVM may be constructed when d is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at s=0 and is closely connected to the Stark conjectures over real quadratic fields. We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact solution to the SIC-POVM problem in dimension 23.