From the Choi Formalism in Infinite Dimensions to Unique Decompositions of Generators of Completely Positive Dynamical Semigroups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916378322141184 |
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| author | Ende, Frederik vom |
| author_facet | Ende, Frederik vom |
| contents | Given any separable complex Hilbert space, any trace-class operator $B$ which does not have purely imaginary trace, and any generator $L$ of a norm-continuous one-parameter semigroup of completely positive maps we prove that there exists a unique bounded operator $K$ and a unique completely positive map $Φ$ such that (i) $L=K(\cdot)+(\cdot)K^*+Φ$, (ii) the superoperator $Φ(B^*(\cdot)B)$ is trace class and has vanishing trace, and (iii) ${\rm tr}(B^*K)$ is a real number. Central to our proof is a modified version of the Choi formalism which relates completely positive maps to positive semi-definite operators. We characterize when this correspondence is injective and surjective, respectively, which in turn explains why the proof idea of our main result cannot extend to non-separable Hilbert spaces. In particular, we find examples of positive semi-definite operators which have empty pre-image under the Choi formalism as soon as the underlying Hilbert space is infinite-dimensional. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_14344 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | From the Choi Formalism in Infinite Dimensions to Unique Decompositions of Generators of Completely Positive Dynamical Semigroups Ende, Frederik vom Functional Analysis Mathematical Physics Quantum Physics 37N20, 46N50, 47B10, 81P48 Given any separable complex Hilbert space, any trace-class operator $B$ which does not have purely imaginary trace, and any generator $L$ of a norm-continuous one-parameter semigroup of completely positive maps we prove that there exists a unique bounded operator $K$ and a unique completely positive map $Φ$ such that (i) $L=K(\cdot)+(\cdot)K^*+Φ$, (ii) the superoperator $Φ(B^*(\cdot)B)$ is trace class and has vanishing trace, and (iii) ${\rm tr}(B^*K)$ is a real number. Central to our proof is a modified version of the Choi formalism which relates completely positive maps to positive semi-definite operators. We characterize when this correspondence is injective and surjective, respectively, which in turn explains why the proof idea of our main result cannot extend to non-separable Hilbert spaces. In particular, we find examples of positive semi-definite operators which have empty pre-image under the Choi formalism as soon as the underlying Hilbert space is infinite-dimensional. |
| title | From the Choi Formalism in Infinite Dimensions to Unique Decompositions of Generators of Completely Positive Dynamical Semigroups |
| topic | Functional Analysis Mathematical Physics Quantum Physics 37N20, 46N50, 47B10, 81P48 |
| url | https://arxiv.org/abs/2401.14344 |