  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.
