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

Author:

Fang KunORCID,Fawzi HamzaORCID

Abstract

AbstractWe 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 $$O(d^2/\ell ^2)$$ O ( d 2 / 2 ) in the regime $$\ell = \Omega (d)$$ = Ω ( d ) where $$\ell $$ 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.

Funder

Isaac Newton Trust

Publisher

Springer Science and Business Media LLC

Subject

General Mathematics,Software

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