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On the Existence of Algebraic Equiangular Lines

Igor Van Loo, Frédérique Oggier

Abstract

We consider real and complex equiangular lines, generated by unit vectors. We show that, for an arbitrary dimension $d$, if there exists a set of $d^2$ equiangular unit vectors in $\mathbb{C}^d$, then there must exist a set of $d^2$ equiangular unit vectors with all of their coefficients in a number field. This result is motivated by the question of constructing SIC-POVMs in quantum physics and conjectures around them. We discuss applications of our techniques to the case of real equiangular lines and consequences of the above results.

On the Existence of Algebraic Equiangular Lines

Abstract

We consider real and complex equiangular lines, generated by unit vectors. We show that, for an arbitrary dimension , if there exists a set of equiangular unit vectors in , then there must exist a set of equiangular unit vectors with all of their coefficients in a number field. This result is motivated by the question of constructing SIC-POVMs in quantum physics and conjectures around them. We discuss applications of our techniques to the case of real equiangular lines and consequences of the above results.
Paper Structure (12 sections, 23 theorems, 59 equations, 1 table)

This paper contains 12 sections, 23 theorems, 59 equations, 1 table.

Table of Contents

  1. Introduction
  2. Reformulating Angle Constraints as a System of Polynomial Equations
  3. Basic polynomial equations for the complex case
  4. Polynomial equations for Weyl-Heisenberg covariant SIC-POVMs
  5. Polynomial equations for the real case
  6. Existence of Algebraic Solutions
  7. Hilbert's Nullstellensatz viewpoint
  8. Gröbner bases viewpoint
  9. Consequences and Discussions
  10. About normalized overlaps
  11. About algebraic Weyl-Heisenberg SIC-POVMs
  12. About real equiangular lines

Key Result

Theorem 3.1

(Weak Nullstellensatz, cox2007) If $K$ is an algebraically closed field and $J \subseteq K[X_1, \ldots, X_n]$ is an ideal such that $\mathbb{V}_K(J) = \varnothing$, then $J = K[X_1, \ldots, X_n]$.

Theorems & Definitions (48)

  • Conjecture 1
  • Conjecture 2
  • Theorem 3.1
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • Corollary 3.5
  • proof
  • ...and 38 more