Noncommutative versions of inequalities in quantum information theory
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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 throw x^α and x^1-α in two 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.
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