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

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

URL: http://arxiv.org/abs/1909.01705v4
Date: Mon, 5 Jun 2023 07:08:36 GMT
Title: A refinement of Reznick's Positivstellensatz with applications to quantum information theory
Authors: Alexander M\"uller-Hermes and Ion Nechita and David Reeb
Abstract summary: In Hilbert's 17th problem Artin showed that any positive definite in several variables can be written as the quotient of two sums of squares. 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.
Score: 72.8349503901712
License: http://creativecommons.org/licenses/by/4.0/
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.

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