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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2411.11035 |
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| _version_ | 1866929600157712384 |
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| author | Pastuszak, Grzegorz Jaworska-Pastuszak, Alicja Kamizawa, Takeo Jamiołkowski, Andrzej |
| author_facet | Pastuszak, Grzegorz Jaworska-Pastuszak, Alicja Kamizawa, Takeo Jamiołkowski, Andrzej |
| contents | A one-parameter family of hermiticity-preserving superoperators is a time-dependent family $\{Φ_{t}\colon\mathbb{M}_{n}(\mathbb{C})\rightarrow\mathbb{M}_{n}(\mathbb{C})\}_{t\in\mathbb{R}}$ 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 $Φ_{t}$ is not positive, $(2)$ $Φ_{t}$ is not positive for $t$ in some open interval $(u,v)\subseteq\mathbb{R}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11035 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On one-parameter families of hermiticity-preserving superoperators which are not positive Pastuszak, Grzegorz Jaworska-Pastuszak, Alicja Kamizawa, Takeo Jamiołkowski, Andrzej Mathematical Physics Logic Operator Algebras Spectral Theory A one-parameter family of hermiticity-preserving superoperators is a time-dependent family $\{Φ_{t}\colon\mathbb{M}_{n}(\mathbb{C})\rightarrow\mathbb{M}_{n}(\mathbb{C})\}_{t\in\mathbb{R}}$ 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 $Φ_{t}$ is not positive, $(2)$ $Φ_{t}$ is not positive for $t$ in some open interval $(u,v)\subseteq\mathbb{R}$ 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. |
| title | On one-parameter families of hermiticity-preserving superoperators which are not positive |
| topic | Mathematical Physics Logic Operator Algebras Spectral Theory |
| url | https://arxiv.org/abs/2411.11035 |