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

Author:

Müller-Hermes Alexander1,Nechita Ion2,Reeb David3

Affiliation:

1. Department 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

2. Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, France

3. Institute for Theoretical Physics, Leibniz Universität Hannover, 30167 Hannover, Germany \ Bosch Center for Artificial Intelligence, Robert-Bosch-Campus 1, 71272 Renningen, Germany

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.

Funder

VILLUM FONDEN

Marie Sklodowska-Curie Action

Agence Nationale de la Recherche

Publisher

Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften

Subject

Physics and Astronomy (miscellaneous),Atomic and Molecular Physics, and Optics

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