Polynomial Optimization on Odd-Dimensional Spheres

D’Angelo, John P.; Putinar, Mihai

doi:10.1007/978-0-387-09686-5_1

Cited by 10 publications

(10 citation statements)

References 29 publications

(16 reference statements)

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“…The converse however is not true. For example p(z,z) = (z +z) 2 is evidently rsos, however it is not csos [16,Example (a)]. Indeed the zero-set of a csos polynomial must be a complex variety, and the zero set of p(z,z) here is the imaginary axis.…”

Section: Hermitian Polynomials and Sums Of Squaresmentioning

confidence: 99%

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

Fang

Fawzi

2020

Math. Program.

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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.

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Section: Hermitian Polynomials and Sums Of Squaresmentioning

confidence: 99%

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

Fang

Fawzi

2020

Math. Program.

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“…and obtain the following conclusions [9,10]: The main result, proved even in a more general context than stated below (for sections of a holomorphic vector bundle on a projective manifold, see [34]) is putting the above observations in their natural higher-dimensional context. Theorem 9.64 (Varolin).…”

Section: Proposition 963 (D'angelo)mentioning

confidence: 91%

“…The third one, which is purely algebraic, is the most accessible for the nonanalyst, but the other two have intrinsic value, as we shall see. For more details and examples the reader can consult the works of d'Angelo and his collaborators [5,6,9,10,11]. Proof 1.…”

Section: Positive Forms In Several Complex Variablesmentioning

confidence: 98%

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Chapter 9: Sums of Hermitian Squares: Old and New

Putinar¹

2012

Semidefinite Optimization and Convex Algebraic Geometry

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This final chapter marks a departure from the main framework of the book by putting emphasis on hermitian forms over the complex field rather than symmetric forms over the real field. The passage is both natural and necessary. To give a simple motivation: polynomial or rational functions with real coefficients, so much praised in the preceding chapters, may very well have complex roots or complex poles. Taking them into account greatly simplifies computations and conceptual thinking, as we all remember from elementary algebra. A second important observation goes back to the dictionary between elementary functions and matrices: by writing in complex coordinates a real valued polynomial (in any number of variables) p(z, z) = c αβ z α z β uniquely determines the hermitian matrix (c αβ ), while a similar decomposition q(x) = γ αβ x α+β , with real coefficients γ αβ , so much needed for semidefinite programming, has a clear ambiguity. The appearance at this late stage of the book of imaginary "ghosts" related to the basic entities encountered so far should not discourage the truly real and very applied reader. IntroductionA question arises from the very beginning: how much of the vast theory of hermitian forms (in a finite or infinite number of variables) should the student or practitioner in applied areas of real algebra, functional analysis, algebraic geometry, or optimization theory know? Due to the depth and wide ramifications of hermitian forms (over the complex field) versus forms over real fields, the answer is: quite a lot! The good news is that the material, old and new, either is well known, circulating in part as folklore, or is accessible, due to a century and a half of continuous development † Mihai Putinar was supported by NSF grant DMS-1001071.

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“…,z n ). This problem is closely related [39] to optimization over the real unit sphere, though not always identical [38]. When d > 1, this is generally NP-hard; see [40].…”

Section: Polynomial Optimization and Sum-of-squares Proofsmentioning

confidence: 99%

Quantum de Finetti Theorems Under Local Measurements with Applications

Brandão

Harrow

2017

Commun. Math. Phys.

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Quantum de Finetti theorems are a useful tool in the study of correlations in quantum multipartite states. In this paper we prove two new quantum de Finetti theorems, both showing that under tests formed by local measurements in each of the subsystems one can get a much improved error dependence on the dimension of the subsystems. We also obtain similar results for non-signaling probability distributions. We give the following applications of the results to quantum complexity theory, polynomial optimization, and quantum information theory:

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Polynomial Optimization on Odd-Dimensional Spheres

Cited by 10 publications

References 29 publications

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

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

Chapter 9: Sums of Hermitian Squares: Old and New

Quantum de Finetti Theorems Under Local Measurements with Applications

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