Partitions of Equiangular Tight Frames

TL;DR

This paper introduces an efficient algorithm for partitioning specific equiangular tight frames that meet the operator norm bounds related to the Kadison-Singer problem, expanding understanding of ETF structures.

Contribution

It provides a new algorithm for partitioning ETFs satisfying the MSS bound and proves bounds for partitions with small largest subsets.

Findings

Partitions of certain ETFs satisfy the MSS bound.

Recursive skew-symmetric conference matrices generate ETFs with desirable partitions.

All partitions with largest subset size three or less meet the MSS bound.

Abstract

We present a new efficient algorithm to construct partitions of a special class of equiangular tight frames (ETFs) that satisfy the operator norm bound established by a theorem of Marcus, Spielman, and Srivastava (MSS), which they proved as a corollary yields a positive solution to the Kadison-Singer problem. In particular, we prove that certain diagonal partitions of complex ETFs generated by recursive skew-symmetric conference matrices yield a refinement of the MSS bound. Moreover, we prove that all partitions of ETFs whose largest subset has cardinality three or less also satisfy the MSS bound.