# Aligned SICs and embedded tight frames in even dimensions

## TL;DR

This paper proves new structural decompositions of aligned SICs in even dimensions into tight frames, extending known results from odd dimensions and exploring parity operator differences in the Clifford group.

## Contribution

It introduces methods to decompose aligned SICs in even dimensions into tight frames, overcoming previous limitations and analyzing parity operator differences.

## Key findings

- Aligned SICs in even dimensions can be partitioned into tight frames.
- Developed new methods applicable to even dimensions.
- Analyzed parity operator differences in the Clifford group.

## Abstract

Alignment is a geometric relation between pairs of Weyl-Heisenberg SICs, one in dimension $d$ and another in dimension $d(d-2)$, manifesting a well-founded conjecture about a number-theoretical connection between the SICs. In this paper, we prove that if $d$ is even, the SIC in dimension $d(d-2)$ of an aligned pair can be partitioned into $(d-2)^2$ tight $d^2$-frames of rank $d(d-1)/2$ and, alternatively, into $d^2$ tight $(d-2)^2$-frames of rank $(d-1)(d-2)/2$. The corresponding result for odd $d$ is already known, but the proof for odd $d$ relies on results which are not available for even $d$. We develop methods that allow us to overcome this issue. In addition, we provide a relatively detailed study of parity operators in the Clifford group, emphasizing differences in the theory of parity operators in even and odd dimensions and discussing consequences due to such differences. In a final section, we study implications of alignment for the symmetry of the SIC.