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

TL;DR

This paper refines Reznick's Positivstellensatz by providing simpler proofs and improvements using quantum information theory concepts, and applies these results to quantum de Finetti theorems and spherical designs.

Contribution

It introduces quantum information techniques to improve bounds and proofs related to Positivstellensatz and explores applications in quantum de Finetti theorems and spherical designs.

Findings

Simpler proofs of Reznick's Positivstellensatz

Minor improvements on bounds for sums of squares representations

Enhanced bounds for exponential quantum de Finetti theorems

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.