A Refinement Of Reznick’s Positivstellensatz With Applications To Quantum Information Theory

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Alexander Müller-Hermes^1,2, Ion Nechita³, and David Reeb^4,5

¹Department of Mathematical Sciences, University of Copenhagen, 2100 Copenhagen, Denmark
²Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex, France
³Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, France
⁴Institute for Theoretical Physics, Leibniz Universität Hannover, 30167 Hannover, Germany
⁵Bosch Center for Artificial Intelligence, Robert-Bosch-Campus 1, 71272 Renningen, Germany

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Abstract

In his solution of Hilbert’s 17th problem Artin showed that any positive definite polynomial in several variables can be written as the quotient of two sums of squares. Later Reznick showed that the denominator in Artin’s result can always be chosen as an $N$-th power of the squared norm of the variables and gave explicit bounds on $N$. By using concepts from quantum information theory (such as partial traces, optimal cloning maps, and an identity due to Chiribella) we give simpler proofs and minor improvements of both real and complex versions of this result. Moreover, we discuss constructions of Hilbert identities using Gaussian integrals and we review an elementary method to construct complex spherical designs. Finally, we apply our results to give improved bounds for exponential quantum de Finetti theorems in the real and in the complex setting.

► BibTeX data

@article{MullerHermes2023refinementof, doi = {10.22331/q-2023-05-12-1001}, url = {https://doi.org/10.22331/q-2023-05-12-1001}, title = {A refinement of {R}eznick’s {P}ositivstellensatz with applications to quantum information theory}, author = {M{“{u}}ller-Hermes, Alexander and Nechita, Ion and Reeb, David}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {1001}, month = may, year = {2023}<br /> }

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