The joint numerical range of three hermitian $4\times 4$ matrices
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917487544631296 |
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| author | Pikul, Piotr Spitkovsky, Ilya Szymański, Konrad Weis, Stephan Życzkowski, Karol |
| author_facet | Pikul, Piotr Spitkovsky, Ilya Szymański, Konrad Weis, Stephan Życzkowski, Karol |
| contents | We analyze the joint numerical range $W$ of three hermitian matrices of order four. In the generic case, this three-dimensional convex set has a smooth boundary. We analyze non-generic structures. Fifteen possible classes regarding the numbers of non-elliptic faces in the boundary of $W$ are identified and an explicit example is presented for each class. Secondly, it is shown that a nonempty intersection of three mutually distinct one-dimensional faces is a corner point. Thirdly, introducing a tensor product structure into $\mathbb C^4=\mathbb C^2\otimes\mathbb C^2$, one defines the separable joint numerical range - a subset of $W$ useful in studies of quantum entanglement. The boundary of the separable numerical range is compared with that of $W$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_27670 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The joint numerical range of three hermitian $4\times 4$ matrices Pikul, Piotr Spitkovsky, Ilya Szymański, Konrad Weis, Stephan Życzkowski, Karol Functional Analysis 15A60, 52A15, 47L07, 52A20 We analyze the joint numerical range $W$ of three hermitian matrices of order four. In the generic case, this three-dimensional convex set has a smooth boundary. We analyze non-generic structures. Fifteen possible classes regarding the numbers of non-elliptic faces in the boundary of $W$ are identified and an explicit example is presented for each class. Secondly, it is shown that a nonempty intersection of three mutually distinct one-dimensional faces is a corner point. Thirdly, introducing a tensor product structure into $\mathbb C^4=\mathbb C^2\otimes\mathbb C^2$, one defines the separable joint numerical range - a subset of $W$ useful in studies of quantum entanglement. The boundary of the separable numerical range is compared with that of $W$. |
| title | The joint numerical range of three hermitian $4\times 4$ matrices |
| topic | Functional Analysis 15A60, 52A15, 47L07, 52A20 |
| url | https://arxiv.org/abs/2510.27670 |