On maps which preserve semipositivity and quantifier elimination theory for real numbers
Grzegorz Pastuszak, Adam Skowyrski, Andrzej Jamiołkowski · Communications in Contemporary Mathematics · 2020
Assume that [Formula: see text] is a superoperator which preserves hermiticity. We give an algorithm determining whether [Formula: see text] preserves semipositivity (we call [Formula: see text] positive in this case). Our approach to the problem has a model-theoretic nature, namely, we apply techniques of quantifier elimination theory for real numbers. An approach based on these techniques seems to be the only one that allows to decide whether an arbitrary hermiticity-preserving [Formula: see text] is positive. Before we go to detailed analysis of the problem, we argue that quantifier elimination for real numbers (and also for complex numbers) can play a significant role in quantum information theory and other areas as well.