Repository logo
 

The sum-of-squares hierarchy on the sphere and applications in quantum information theory

Published version
Peer-reviewed

Change log

Abstract

Abstract: We consider the problem of maximizing a homogeneous polynomial on the unit sphere and its hierarchy of sum-of-squares relaxations. Exploiting the polynomial kernel technique, we obtain a quadratic improvement of the known convergence rate by Reznick and Doherty and Wehner. Specifically, we show that the rate of convergence is no worse than O(d2/ℓ2) in the regime ℓ=Ω(d) where ℓ is the level of the hierarchy and d the dimension, solving a problem left open in the recent paper by de Klerk and Laurent (arXiv:1904.08828 ). Importantly, our analysis also works for matrix-valued polynomials on the sphere which has applications in quantum information for the Best Separable State problem. By exploiting the duality relation between sums of squares and the Doherty–Parrilo–Spedalieri hierarchy in quantum information theory, we show that our result generalizes to nonquadratic polynomials the convergence rates of Navascués, Owari and Plenio.

Description

Keywords

Full Length Paper, 90C22, 90C23, 81P42

Journal Title

Mathematical Programming

Conference Name

Journal ISSN

0025-5610
1436-4646

Volume Title

190

Publisher

Springer Berlin Heidelberg
Sponsorship
Isaac Newton Trust (RG74916)