Noncommutative versions of inequalities in quantum information theory
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Abstract
In this paper, we aim to replace in the definitions of covariance and correlation the usual trace 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.
Keywords
Tracial positive linear map Wigner–Yanase skew information Covariance Correlation Uncertainty relationMathematics Subject Classification
Primary 46L05 47A63 Secondary 81P15Notes
Acknowledgements
The second author (corresponding author) is supported by a grant from the Iran National Science Foundation (96006713). The third author is partially supported by JPSP KAKENHI Grant Number 19K03525.
Compliance with ethical standards
Conflict of interest
The authors declare no conflict of interest.
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