The sum-of-squares hierarchy on the sphere and applications in quantum information theory
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jats:titleAbstract</jats:title>jats:pWe consider the problem of maximizing a homogeneous polynomial on the unit sphere and its hierarchy of sum-of-squares relaxations. Exploiting the jats:italicpolynomial kernel technique</jats:italic>, 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 jats:inline-formulajats:alternativesjats:tex-math$$O(d^2/\ell ^2)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miO</mml:mi> mml:mo(</mml:mo> mml:msup mml:mid</mml:mi> mml:mn2</mml:mn> </mml:msup> mml:mo/</mml:mo> mml:msup mml:miℓ</mml:mi> mml:mn2</mml:mn> </mml:msup> mml:mo)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> in the regime jats:inline-formulajats:alternativesjats:tex-math$$\ell = \Omega (d)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miℓ</mml:mi> mml:mo=</mml:mo> mml:miΩ</mml:mi> mml:mo(</mml:mo> mml:mid</mml:mi> mml:mo)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> where jats:inline-formulajats:alternativesjats:tex-math$$\ell $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miℓ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> is the level of the hierarchy and jats:italicd</jats:italic> the dimension, solving a problem left open in the recent paper by de Klerk and Laurent (<jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" ext-link-type="uri" xlink:href="http://arxiv.org/abs/1904.08828">arXiv:1904.08828</jats:ext-link> ). 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.</jats:p>
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1436-4646