The Paulsen Problem Made Simple

TL;DR

This paper simplifies the proof of the Paulsen problem in operator theory, providing a more accessible approach and improving the bound on the distance to an equal norm Parseval frame.

Contribution

It introduces a simpler proof using radial isotropic position and improves the bound from $O(\epsilon d^{13/2})$ to $O(\epsilon d^2)$.

Findings

Simplified proof of the Paulsen problem.

Improved bound on the distance to an equal norm Parseval frame.

Enhanced understanding of operator theory concepts.

Abstract

The Paulsen problem is a basic problem in operator theory that was resolved in a recent tour-de-force work of Kwok, Lau, Lee and Ramachandran. In particular, they showed that every $\epsilon$-nearly equal norm Parseval frame in $d$ dimensions is within squared distance $O(\epsilon d^{13/2})$ of an equal norm Parseval frame. We give a dramatically simpler proof based on the notion of radial isotropic position, and along the way show an improved bound of $O(\epsilon d^2)$.