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

• Published in: Mathematical programming Vol. 190; no. 1-2; pp. 331 - 360
• Main Authors: Fang, Kun, Fawzi, Hamza
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2021; Springer Nature B.V
• Subjects: Calculus of Variations and Optimal Control; Optimization; Combinatorics; Convergence; Full Length Paper; Information theory; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Polynomials; Problem solving; Quantum phenomena; Theoretical; 81P42; 90C22; 90C23
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-020-01537-7

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 ( d 2 / ℓ 2 ) 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.