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Article Dans Une Revue Quantum Année : 2023

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

Résumé

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 a linear form 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 de Finetti theorems in the real and in the complex setting.

Dates et versions

hal-02988627 , version 1 (04-11-2020)

Identifiants

Citer

Alexander Müller-Hermes, Ion Nechita, David Reeb. A refinement of Reznick's Positivstellensatz with applications to quantum information theory. Quantum, 2023, 7, pp.1001. ⟨10.22331/q-2023-05-12-1001⟩. ⟨hal-02988627⟩
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