Fixed Points of Completely Positive Trace-Preserving Maps in Infinite Dimension
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| Format: | Preprint |
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2024
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| _version_ | 1866917844972732416 |
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| author | Tumulka, Roderich Weixler, Jonte |
| author_facet | Tumulka, Roderich Weixler, Jonte |
| contents | Completely positive trace-preserving maps $S$, also known as quantum channels, arise in quantum physics as a description of how the density operator $ρ$ of a system changes in a given time interval, allowing not only for unitary evolution but arbitrary operations including measurements or other interaction with an environment. It is known that if the Hilbert space $\mathscr{H}$ that $ρ$ acts on is finite-dimensional, then every $S$ must have a fixed point, i.e., a density operator $ρ_0$ with $S(ρ_0)=ρ_0$. In infinite dimension, $S$ need not have a fixed point in general. However, we prove here the existence of a fixed point under a certain additional assumption which is, roughly speaking, that $S$ leaves invariant a certain set of density operators with bounded ``cost'' of preparation. The proof is an application of the Schauder-Tychonoff fixed point theorem. Our motivation for this question comes from a proposal of Deutsch for how to define quantum theory in a space-time with closed timelike curves; our result supports the viability of Deutsch's proposal. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_14800 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fixed Points of Completely Positive Trace-Preserving Maps in Infinite Dimension Tumulka, Roderich Weixler, Jonte Mathematical Physics Quantum Physics Completely positive trace-preserving maps $S$, also known as quantum channels, arise in quantum physics as a description of how the density operator $ρ$ of a system changes in a given time interval, allowing not only for unitary evolution but arbitrary operations including measurements or other interaction with an environment. It is known that if the Hilbert space $\mathscr{H}$ that $ρ$ acts on is finite-dimensional, then every $S$ must have a fixed point, i.e., a density operator $ρ_0$ with $S(ρ_0)=ρ_0$. In infinite dimension, $S$ need not have a fixed point in general. However, we prove here the existence of a fixed point under a certain additional assumption which is, roughly speaking, that $S$ leaves invariant a certain set of density operators with bounded ``cost'' of preparation. The proof is an application of the Schauder-Tychonoff fixed point theorem. Our motivation for this question comes from a proposal of Deutsch for how to define quantum theory in a space-time with closed timelike curves; our result supports the viability of Deutsch's proposal. |
| title | Fixed Points of Completely Positive Trace-Preserving Maps in Infinite Dimension |
| topic | Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2411.14800 |