  Assume that
$\Phi:\mathbb{M}_{n}(\mathbb{C})\rightarrow\mathbb{M}_{n}(\mathbb{C})$ is a
superoperator which preserves hermiticity. We give an algorithm determining
whether $\Phi$ preserves semipositivity (we call $\Phi$ 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 $\Phi$ 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.
