Decomposition of tracial positive maps and applications in quantum information
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arXiv
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| Format: | Preprint |
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2022
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| author | Dadkha, Ali Kian, Mohsen Moslehian, Mohammad Sal |
| author_facet | Dadkha, Ali Kian, Mohsen Moslehian, Mohammad Sal |
| contents | Every positive multilinear map between $C^*$-algebras is separately weak$^*$-continuous. We show that the joint weak$^*$-continuity is equivalent to the joint weak$^*$-continuity of the multiplications of $C^*$-algebras under consideration. We study the behavior of general tracial positive maps on properly infinite von Neumann algebras and by applying the Aron--Berner extension of multilinear maps, we establish that under some mild conditions every tracial positive multilinear map between general $C^*$-algebras enjoys a decomposition $Φ=φ_2 \circ φ_1$, in which $φ_1$ is a tracial positive linear map with the commutative range and $φ_2$ is a tracial completely positive map with the commutative domain. As an immediate consequence, tracial positive multilinear maps are completely positive. Furthermore, we prove that if the domain of a general tracial completely positive map $Φ$ between $C^*$-algebra is a von Neumann algebra, then $Φ$ has a similar decomposition. As an application, we investigate the generalized variance and covariance in quantum mechanics via arbitrary positive maps. Among others, an uncertainty relation inequality for commuting observables in a composite physical system is presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_12798 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Decomposition of tracial positive maps and applications in quantum information Dadkha, Ali Kian, Mohsen Moslehian, Mohammad Sal Operator Algebras Functional Analysis Primary 47C15, 46L10, Secondary 47A63, 46L05 Every positive multilinear map between $C^*$-algebras is separately weak$^*$-continuous. We show that the joint weak$^*$-continuity is equivalent to the joint weak$^*$-continuity of the multiplications of $C^*$-algebras under consideration. We study the behavior of general tracial positive maps on properly infinite von Neumann algebras and by applying the Aron--Berner extension of multilinear maps, we establish that under some mild conditions every tracial positive multilinear map between general $C^*$-algebras enjoys a decomposition $Φ=φ_2 \circ φ_1$, in which $φ_1$ is a tracial positive linear map with the commutative range and $φ_2$ is a tracial completely positive map with the commutative domain. As an immediate consequence, tracial positive multilinear maps are completely positive. Furthermore, we prove that if the domain of a general tracial completely positive map $Φ$ between $C^*$-algebra is a von Neumann algebra, then $Φ$ has a similar decomposition. As an application, we investigate the generalized variance and covariance in quantum mechanics via arbitrary positive maps. Among others, an uncertainty relation inequality for commuting observables in a composite physical system is presented. |
| title | Decomposition of tracial positive maps and applications in quantum information |
| topic | Operator Algebras Functional Analysis Primary 47C15, 46L10, Secondary 47A63, 46L05 |
| url | https://arxiv.org/abs/2202.12798 |