CP$^{\infty}$ and beyond: 2-categorical dilation theory
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866912138207952896 |
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| author | Allen, Robert Verdon, Dominic |
| author_facet | Allen, Robert Verdon, Dominic |
| contents | The problem of extending the insights and techniques of categorical quantum mechanics to infinite-dimensional systems was considered in (Coecke and Heunen, 2016). In that work the $\mathrm{CP}^{\infty}$-construction, which recovers the category of Hilbert spaces and quantum operations from the category of Hilbert spaces and bounded linear maps, was defined. Here we show that by a `horizontal categorification' of the $\mathrm{CP}^{\infty}$-construction, one can recover the category of all von Neumann algebras and channels (normal unital completely positive maps) from the 2-category $[W^*]$ of von Neumann algebras, bimodules and intertwiners. As an application, we extend Choi's characterisation of extremal channels between finite-dimensional matrix algebras to a characterisation of extremal channels between arbitrary von Neumann algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_15776 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | CP$^{\infty}$ and beyond: 2-categorical dilation theory Allen, Robert Verdon, Dominic Operator Algebras Category Theory Quantum Physics The problem of extending the insights and techniques of categorical quantum mechanics to infinite-dimensional systems was considered in (Coecke and Heunen, 2016). In that work the $\mathrm{CP}^{\infty}$-construction, which recovers the category of Hilbert spaces and quantum operations from the category of Hilbert spaces and bounded linear maps, was defined. Here we show that by a `horizontal categorification' of the $\mathrm{CP}^{\infty}$-construction, one can recover the category of all von Neumann algebras and channels (normal unital completely positive maps) from the 2-category $[W^*]$ of von Neumann algebras, bimodules and intertwiners. As an application, we extend Choi's characterisation of extremal channels between finite-dimensional matrix algebras to a characterisation of extremal channels between arbitrary von Neumann algebras. |
| title | CP$^{\infty}$ and beyond: 2-categorical dilation theory |
| topic | Operator Algebras Category Theory Quantum Physics |
| url | https://arxiv.org/abs/2310.15776 |