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

We consider the problem of maximizing a homogeneous polynomial on the unit sphere and its hierarchy of Sum-of-Squares (SOS) relaxations. Exploiting the polynomial kernel technique, we obtain a quadratic improvement of the known convergence rate by Reznick and Doherty & Wehner. Specifically, we show that the rate of convergence is no worse than $O(d^2/\ell^2)$ in the regime $\ell \geq \Omega(d)$ where $\ell$ is the level of the hierarchy and $d$ the dimension, solving a problem left open in the recent paper by de Klerk & 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 DPS hierarchy in quantum information theory, we show that our result generalizes to nonquadratic polynomials the convergence rates of Navascu\'es, Owari & Plenio.