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

2024/11/17 by Grzegorz Pastuszak, Alicja Jaworska-Pastuszak, Pastuszak, Grzegorz +5 · 2 citations
Computer Science · Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Logic (math.LO) #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Operator Algebras (math.OA) #Polynomial and algebraic computation #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2411.11035

openalex publication_date 2024/11/17 · openalex created_date 2024/11/22 · openalex updated_date 2026/07/28

Abstract

A one-parameter family of hermiticity-preserving superoperators is a time-dependent family \Φ_t\colon\mathbbM_n(ℂ)→\mathbbM_n(ℂ)\_t∈ℝ of hermiticity-preserving superoperators determined, in a certain sense, by real and complex polynomial functions in the variable t∈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 Φ_t is not positive, (2) Φ_t is not positive for t in some open interval (u,v)⊆ℝ and (3) there is some Φ_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.

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