Linear Dependencies in Weyl-Heisenberg Orbits

TL;DR

This paper investigates linear dependencies in Weyl-Heisenberg SICs across various dimensions, revealing structural patterns linked to algebraic geometry and implications for the SIC existence problem.

Contribution

It proves that linear dependencies always occur in Weyl-Heisenberg orbits under specific conditions and explores their implications for SIC structures and algebraic geometry connections.

Findings

Linear dependencies emerge in Weyl-Heisenberg orbits when the fiducial vector is in certain subspaces.

Dimension 6 SICs contain smaller SIC structures, affecting the SIC existence problem.

Results extend to orbits with fiducial vectors in eigenspaces of other Clifford group elements.

Abstract

Five years ago, Lane Hughston showed that some of the symmetric informationally complete positive operator valued measures (SICs) in dimension 3 coincide with the Hesse configuration (a structure well known to algebraic geometers, which arises from the torsion points of a certain elliptic curve). This connection with elliptic curves is signalled by the presence of linear dependencies among the SIC vectors. Here we look for analogous connections between SICs and algebraic geometry by performing computer searches for linear dependencies in higher dimensional SICs. We prove that linear dependencies will always emerge in Weyl-Heisenberg orbits when the fiducial vector lies in a certain subspace of an order 3 unitary matrix. This includes SICs when the dimension is divisible by 3 or equal to 8 mod 9. We examine the linear dependencies in dimension 6 in detail and show that smaller dimensional SICs are contained within this structure, potentially impacting the SIC existence problem. We extend our results to look for linear dependencies in orbits when the fiducial vector lies in an eigenspace of other elements of the Clifford group that are not order 3. Finally, we align our work with recent studies on representations of the Clifford group.