A refinement of Reznick’s Positivstellensatz with applications to quantum information theory

Alexander Müller-Hermes1, Ion Nechita2, and David Reeb3

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

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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.

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