Ergodic Quantum Processes on Finite von Neumann Algebras
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909681926012928 |
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| author | Nelson, Brent Roon, Eric B. |
| author_facet | Nelson, Brent Roon, Eric B. |
| contents | Let $(M,τ)$ be a tracial von Neumann algebra with a separable predual and let $(Ω, \mathbb{P})$ be a probability space. A bounded positive random linear operator on $L^1(M,τ)$ is a map $γ: Ω\times L^1(M,τ) \to L^1(M,τ)$ so that $τ(γ_ω(x)a)$ is measurable for all $x\in L^1(M,τ)$ and $a\in M$, and $x\mapsto γ_ω(x)$ is bounded, positive, and linear almost surely. Given an ergodic $T\in Aut(Ω, \mathbb{P})$, we study quantum processes of the form $γ_{T^n ω}\circ γ_{T^{n-1}ω} \circ \cdots \circ γ_{T^mω}$ for $m,n\in \mathbb{Z}$. Using the Hennion metric introduced in [MS22], we show that under reasonable assumptions such processes collapse to replacement channels exponentially fast almost surely. Of particular interest is the case when $γ_ω$ is the predual of a normal positive linear map on $M$. As an example application, we study the clustering properties of normal states that are generated by such random linear operators. These results offer an infinite dimensional generalization of the theorems in [MS22]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_03363 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ergodic Quantum Processes on Finite von Neumann Algebras Nelson, Brent Roon, Eric B. Operator Algebras Mathematical Physics Let $(M,τ)$ be a tracial von Neumann algebra with a separable predual and let $(Ω, \mathbb{P})$ be a probability space. A bounded positive random linear operator on $L^1(M,τ)$ is a map $γ: Ω\times L^1(M,τ) \to L^1(M,τ)$ so that $τ(γ_ω(x)a)$ is measurable for all $x\in L^1(M,τ)$ and $a\in M$, and $x\mapsto γ_ω(x)$ is bounded, positive, and linear almost surely. Given an ergodic $T\in Aut(Ω, \mathbb{P})$, we study quantum processes of the form $γ_{T^n ω}\circ γ_{T^{n-1}ω} \circ \cdots \circ γ_{T^mω}$ for $m,n\in \mathbb{Z}$. Using the Hennion metric introduced in [MS22], we show that under reasonable assumptions such processes collapse to replacement channels exponentially fast almost surely. Of particular interest is the case when $γ_ω$ is the predual of a normal positive linear map on $M$. As an example application, we study the clustering properties of normal states that are generated by such random linear operators. These results offer an infinite dimensional generalization of the theorems in [MS22]. |
| title | Ergodic Quantum Processes on Finite von Neumann Algebras |
| topic | Operator Algebras Mathematical Physics |
| url | https://arxiv.org/abs/2309.03363 |