On One-Parameter Families of Hermiticity-Preserving Superoperators which Are Not Positive

A one-parameter family of hermiticity-preserving superoperators is a time-dependent family [Formula: see text] of hermiticity-preserving superoperators determined, in a certain sense, by real and complex polynomial functions in the variable [Formula: see text]. The paper studies sufficient computable criteria for nonpositivity of superoperators in one-parameter families. More precisely, we give sufficient conditions for the following assertions to hold: [Formula: see text] every [Formula: see text] is not positive, [Formula: see text] [Formula: see text] is not positive for [Formula: see text] in some [Formula: see text] and [Formula: see text] there is some [Formula: see text] which is not positive. We show that [Formula: see text] implies [Formula: see text]. Our approach to the problem is based on the Descartes rule of signs and the Sturm-Tarski theorem. In order to apply these facts, we introduce the sign variation formulas. These formulas are first order logical formulas in one variable [Formula: see text], generalising sign sequences of polynomials used in Descartes rule of signs.