Dimension towers of SICs. I. Aligned SICs and embedded tight frames

TL;DR

This paper explores the relationship between SIC-POVMs in different dimensions, introducing the concept of aligned SICs, providing examples, conjecturing their existence, and proving properties related to entanglement and mutually unbiased bases.

Contribution

It defines aligned SICs across dimensions, provides examples, and proves properties about their structure and relation to mutually unbiased bases.

Findings

19 examples of aligned SICs are provided.

Aligned SICs contain embedded tight frames.

For odd dimensions, vectors have a special entanglement structure.

Abstract

Algebraic number theory relates SIC-POVMs in dimension $d>3$ to those in dimension $d(d-2)$. We define a SIC in dimension $d(d-2)$ to be aligned to a SIC in dimension $d$ if and only if the squares of the overlap phases in dimension $d$ appear as a subset of the overlap phases in dimension $d(d-2)$ in a specified way. We give 19 (mostly numerical) examples of aligned SICs. We conjecture that given any SIC in dimension $d$ there exists an aligned SIC in dimension $d(d-2)$. In all our examples the aligned SIC has lower dimensional equiangular tight frames embedded in it. If $d$ is odd so that a natural tensor product structure exists, we prove that the individual vectors in the aligned SIC have a very special entanglement structure, and the existence of the embedded tight frames follows as a theorem. If $d-2$ is an odd prime number we prove that a complete set of mutually unbiased bases can be obtained by reducing an aligned SIC to this dimension.