Understanding and Generalizing Unique Decompositions of Generators of Dynamical Semigroups
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911941409112064 |
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| author | Ende, Frederik vom |
| author_facet | Ende, Frederik vom |
| contents | We generalize the result of Gorini, Kossakowski, and Sudarshan [J. Math. Phys. 17:821, 1976] that every generator of a quantum-dynamical semigroup decomposes uniquely into a closed and a dissipative part, assuming the trace of both vanishes. More precisely, we show that given any generator $L$ of a completely positive dynamical semigroup and any matrix $B$ there exists a unique matrix $K$ and a unique completely positive map $Φ$ such that (i) $L=K(\cdot)+(\cdot)K^*+Φ$, (ii) the superoperator $Φ(B^*(\cdot)B)$ has trace zero, and (iii) ${\rm tr}(B^*K)$ is a real number. The key to proving this is the relation between the trace of a completely positive map, the trace of its Kraus operators, and expectation values of its Choi matrix. Moreover, we show that the above decomposition is orthogonal with respect to some $B$-weighted inner product. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_04037 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Understanding and Generalizing Unique Decompositions of Generators of Dynamical Semigroups Ende, Frederik vom Quantum Physics Mathematical Physics We generalize the result of Gorini, Kossakowski, and Sudarshan [J. Math. Phys. 17:821, 1976] that every generator of a quantum-dynamical semigroup decomposes uniquely into a closed and a dissipative part, assuming the trace of both vanishes. More precisely, we show that given any generator $L$ of a completely positive dynamical semigroup and any matrix $B$ there exists a unique matrix $K$ and a unique completely positive map $Φ$ such that (i) $L=K(\cdot)+(\cdot)K^*+Φ$, (ii) the superoperator $Φ(B^*(\cdot)B)$ has trace zero, and (iii) ${\rm tr}(B^*K)$ is a real number. The key to proving this is the relation between the trace of a completely positive map, the trace of its Kraus operators, and expectation values of its Choi matrix. Moreover, we show that the above decomposition is orthogonal with respect to some $B$-weighted inner product. |
| title | Understanding and Generalizing Unique Decompositions of Generators of Dynamical Semigroups |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2310.04037 |