Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Conjecture on Almost Flat SIC-POVMs

Ingemar Bengtsson, Markus Grassl

TL;DR

The work investigates whether the X-overlap equation, derived from an almost-flat fiducial Ansatz for SIC-POVMs invariant under anti-unitary symmetry, suffices to enforce full SIC conditions. Using Gröbner-basis methods across several dimensions (notably $d=7,19,67,39,199$) and introducing Legendre vectors, the authors show that the X-overlap identity alone admits non-SIC solutions, including Legendre-type vectors in many primes $d eq3,7,19$, demonstrating that the obstruction to SIC-uniqueness is mild but real. The results illuminate deep connections between SIC-overlaps, Stark-unit structures in ray class fields, and permutation symmetries, while highlighting that the X-overlap relation is not in general sufficient to fix a SIC fiducial. The findings motivate further study into the precise conditions under which the X-overlap identity, together with symmetry and field-theoretic constraints, can determine SIC fiducials or at least significantly constrain their structure.

Abstract

A well supported conjecture states that SIC-POVMs -- maximal sets of complex equiangular lines -- with anti-unitary symmetry give rise to an identity expressing some of its overlaps as squares of the (rescaled) components of a suitably chosen fiducial vector. In number theoretical terms the identity essentially expresses Stark units as sums of products of pairs of square roots of Stark units. We investigate whether the identity is enough to determine these Stark units. The answer is no, but the failure might be quite mild.

A Conjecture on Almost Flat SIC-POVMs

TL;DR

The work investigates whether the X-overlap equation, derived from an almost-flat fiducial Ansatz for SIC-POVMs invariant under anti-unitary symmetry, suffices to enforce full SIC conditions. Using Gröbner-basis methods across several dimensions (notably ) and introducing Legendre vectors, the authors show that the X-overlap identity alone admits non-SIC solutions, including Legendre-type vectors in many primes , demonstrating that the obstruction to SIC-uniqueness is mild but real. The results illuminate deep connections between SIC-overlaps, Stark-unit structures in ray class fields, and permutation symmetries, while highlighting that the X-overlap relation is not in general sufficient to fix a SIC fiducial. The findings motivate further study into the precise conditions under which the X-overlap identity, together with symmetry and field-theoretic constraints, can determine SIC fiducials or at least significantly constrain their structure.

Abstract

A well supported conjecture states that SIC-POVMs -- maximal sets of complex equiangular lines -- with anti-unitary symmetry give rise to an identity expressing some of its overlaps as squares of the (rescaled) components of a suitably chosen fiducial vector. In number theoretical terms the identity essentially expresses Stark units as sums of products of pairs of square roots of Stark units. We investigate whether the identity is enough to determine these Stark units. The answer is no, but the failure might be quite mild.

Paper Structure

This paper contains 12 sections, 3 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A closer look at the Ansatz and the $X$-overlap equation
  3. Gröbner basis calculations
  4. Dimension $7$
  5. Dimension $19$
  6. Dimension $67$
  7. Further results
  8. Dimension $39$
  9. Dimension $199$
  10. Even dimensions
  11. Legendre vectors
  12. Conclusions

Key Result

Proposition 1

Let $\psi_i$ denote the components of a vector $|\Psi\rangle$ that obeys the Ansatz given in eqs. eq:Ansatz1 and Ansatz2. Then for $j\ne 0$

Figures (1)

  • Figure 1: Following Perron, we illustrate our use of his theorems for $d = 11$. Reste are marked with bullets or in the exceptional case $0$ with a ring, while Nichreste are marked with small dots. Shifting the outer rim by an integer we find that the Reste line up in $(d+1)/4$ cases, while Nichtreste line up in $(d-3)/4$ cases. However, for our argument it is important that $0$ is treated separately. It lines up with a Rest if the shift is a quadratic residue (like $1$), and with a Nichtrest if the shift is a non-residue (like $2$).

Theorems & Definitions (3)

  • Proposition 1
  • Lemma 1
  • Proposition 2