Optimal Parseval frames: Total coherence and total volume

TL;DR

This paper introduces new measures for Parseval frames and proves that equiangular Parseval frames optimize these measures, extending classical bounds and raising open questions.

Contribution

It establishes that equiangular Parseval frames maximize total coherence, total volume, and nuclear energy, and generalizes the Welch bound for equal norm frames.

Findings

Equiangular Parseval frames maximize total coherence, total volume, and nuclear energy.

A generalized Welch bound for the smallest k-dimensional volume in equal norm frames.

Derivation of bounds on frame volumes and open questions for future research.

Abstract

We introduce three quantities called total coherence, total volume, and nuclear energy, and we show that equiangular Parseval frames maximize all three of these quantities over the set of all Parseval frames. We then show that equiangular Parseval frames also maximize the total volume and nuclear energy over the set of equal norm frames. Along the way we derive a bound on the smallest k-dimensional volume in an equal norm frame which is a generalization of the Welch bound. We conclude with an extensive list of open questions.