# Constructing exact symmetric informationally complete measurements from numerical solutions

## Marcus Appleby, Tuan-Yow Chien, Steven Flammia and Shayne Waldron

## Abstract:

Recently, several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements (SIC POVMs, or SICs) and algebraic number theory. These conjectures relate the SICs and their minimal defining algebraic number field. Testing or sharpening these conjectures requires that the SICs are expressed exactly, rather than as numerical approximations. While many exact solutions of SICs have been constructed previously using Gr\"obner bases, this method has probably been taken as far as is possible with current computer technology. Here we describe a method for converting high-precision numerical solutions into exact ones using an integer relation algorithm in conjunction with the Galois symmetries of a SIC. Using this method we have calculated 69 new exact solutions, including 9 new dimensions where previously only numerical solutions were known, which more than triples the number of known exact solutions. In some cases the solutions require number fields with degrees as high as 12,288. We use these solutions to confirm that they obey the number-theoretic conjectures and we address two questions suggested by the previous work.

**Keywords:**
SIC,
Zauner's conjecture

**Math Review Classification:**
Primary 05B30, 42C15, 65D30, 81P15;
Secondary 51F25, 81R05

**Length:** 20 pages + 19 page appendix with many data tables

**Last Updated:** 21 March 2017

## Availability: