On one-parameter families of hermiticity-preserving superoperators which are not positive

TL;DR

This paper develops criteria to determine when a one-parameter family of hermiticity-preserving superoperators fails to be positive, using polynomial sign variation methods like Descartes rule of signs and Sturm-Tarski theorem.

Contribution

It introduces computable sign variation formulas to analyze nonpositivity in parameterized superoperator families, extending existing mathematical tools.

Findings

Provides sufficient conditions for superoperators to be nonpositive

Establishes links between nonpositivity at a point and over intervals

Introduces sign variation formulas for polynomial analysis

Abstract

A one-parameter family of hermiticity-preserving superoperators is a time-dependent family $\{\Phi_{t}\colon\mathbb{M}_{n}(\mathbb{C})\rightarrow\mathbb{M}_{n}(\mathbb{C})\}_{t\in\mathbb{R}}$ of hermiticity-preserving superoperators determined, in a certain sense, by real and complex polynomial functions in the variable $t\in\mathbb{R}$. 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: $(1)$ every $\Phi_{t}$ is not positive, $(2)$ $\Phi_{t}$ is not positive for $t$ in some open interval $(u,v)\subseteq\mathbb{R}$ and $(3)$ there is some $\Phi_{t}$ which is not positive. We show that in some situations $(3)$ implies $(2)$. 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 free variable $t$, generalising sign sequences of polynomials used in Descartes rule of signs.