Quantum Coherence Spaces Revisited: A von Neumann (Co)Algebraic Approach
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866912840877604864 |
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| author | Li, Thea Zamdzhiev, Vladimir |
| author_facet | Li, Thea Zamdzhiev, Vladimir |
| contents | We describe a categorical model of MALL (Multiplicative Additive Linear Logic) inspired by the Heisenberg-Schrödinger duality of finite-dimensional quantum theory. Proofs of formulas with positive logical polarity correspond to CPTP (completely positive trace-preserving) maps in our model, i.e. the quantum operations in the Schrödinger picture, whereas proofs of formulas with negative logical polarity correspond to CPU (completely positive unital) maps, i.e. the quantum operations in the Heisenberg picture. The mathematical development is based on noncommutative geometry and finite-dimensional von Neumann (co)algebras, which can be defined as special kinds of (co)monoid objects internal to the category of finite-dimensional operator spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15832 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantum Coherence Spaces Revisited: A von Neumann (Co)Algebraic Approach Li, Thea Zamdzhiev, Vladimir Category Theory Logic in Computer Science Functional Analysis Operator Algebras Quantum Physics We describe a categorical model of MALL (Multiplicative Additive Linear Logic) inspired by the Heisenberg-Schrödinger duality of finite-dimensional quantum theory. Proofs of formulas with positive logical polarity correspond to CPTP (completely positive trace-preserving) maps in our model, i.e. the quantum operations in the Schrödinger picture, whereas proofs of formulas with negative logical polarity correspond to CPU (completely positive unital) maps, i.e. the quantum operations in the Heisenberg picture. The mathematical development is based on noncommutative geometry and finite-dimensional von Neumann (co)algebras, which can be defined as special kinds of (co)monoid objects internal to the category of finite-dimensional operator spaces. |
| title | Quantum Coherence Spaces Revisited: A von Neumann (Co)Algebraic Approach |
| topic | Category Theory Logic in Computer Science Functional Analysis Operator Algebras Quantum Physics |
| url | https://arxiv.org/abs/2601.15832 |