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

中文翻译：

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球面上的平方和层次及其在量子信息论中的应用

我们考虑最大化单位球面上的齐次多项式及其平方和松弛层次的问题。利用多项式核技术，我们获得了 Reznick、Doherty 和 Wehner 已知收敛速度的二次改进。具体来说，我们表明收敛速度不比 $$O(d^2/\ell ^2)$$ O ( d 2 / ℓ 2 ) 差 $$\ell = \Omega (d)$ $ ℓ = Ω ( d ) 其中 $$\ell $$ ℓ 是层次结构的级别，d 是维度，解决了 de Klerk 和 Laurent 近期论文 (arXiv:1904.08828) 中未解决的问题。重要的是，我们的分析也适用于球体上的矩阵值多项式，该球体在量子信息中可应用于最佳可分离状态问题。