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Noncommutative versions of inequalities in quantum information theory
- Source :
- Anal. Math. Phys. 9 (2019), no. 4, 2151--2169
- Publication Year :
- 2019
-
Abstract
- In this paper, we aim to replace in the definitions of covariance and correlation the usual trace {\rm Tr} by a tracial positive map between unital $C^*$-algebras and to replace the functions $x^{\alpha}$ and $x^{1-\alpha}$ by functions $f$ and $g$ satisfying some mild conditions. These allow us to define the generalized covariance, the generalized variance, the generalized correlation and the generalized Wigner--Yanase--Dyson skew information related to the tracial positive maps and functions $f$ and $g$. We persent a generalization of Heisenberg's uncertainty relation in the noncommutative framework. We extend some inequalities and properties for the generalized correlation and the generalized Wigner--Yanase--Dyson skew information. Furthermore, we extend some inequalities for the generalized skew information such as uncertainty relation and the relation between the generalized variance and the generalized skew information.<br />Comment: 19 pages, to appear in Analysis and Mathematical Physics. Some corrections has been done
Details
- Database :
- arXiv
- Journal :
- Anal. Math. Phys. 9 (2019), no. 4, 2151--2169
- Publication Type :
- Report
- Accession number :
- edsarx.1905.02014
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1007/s13324-019-00309-7