Characterizations of homomorphisms among unital completely positive maps
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917895906263040 |
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| author | Kornell, Andre |
| author_facet | Kornell, Andre |
| contents | We prove that a unital completely positive map between finite-dimensional C*-algebras is a homomorphism if and only if it is completely entropy-nonincreasing, where the relevant notion of entropy is a variant of von Neumann entropy. This adjusted von Neumann entropy is the negative of the relative entropy with respect to the uniform state on the C*-algebra, up to an additive constant. As an intermediate step, we prove that a unital completely positive map between finite-dimensional C*-algebras is a homomorphism if and only if its adjusted Choi operator is a projection. Both equivalences generalize familiar facts about stochastic maps between finite sets. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_07229 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterizations of homomorphisms among unital completely positive maps Kornell, Andre Operator Algebras Mathematical Physics Functional Analysis Quantum Physics 46L07 (Primary) 46L30, 94A17 (Secondary) We prove that a unital completely positive map between finite-dimensional C*-algebras is a homomorphism if and only if it is completely entropy-nonincreasing, where the relevant notion of entropy is a variant of von Neumann entropy. This adjusted von Neumann entropy is the negative of the relative entropy with respect to the uniform state on the C*-algebra, up to an additive constant. As an intermediate step, we prove that a unital completely positive map between finite-dimensional C*-algebras is a homomorphism if and only if its adjusted Choi operator is a projection. Both equivalences generalize familiar facts about stochastic maps between finite sets. |
| title | Characterizations of homomorphisms among unital completely positive maps |
| topic | Operator Algebras Mathematical Physics Functional Analysis Quantum Physics 46L07 (Primary) 46L30, 94A17 (Secondary) |
| url | https://arxiv.org/abs/2403.07229 |