Analysis and Mathematical Physics, Year (2019-5)

#### Title : ( 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 {\\\\\\\\\\\\\\\\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 present 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 relation
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@article{paperid:1074157,
title = {Noncommutative versions of inequalities in quantum information theory},
journal = {Analysis and Mathematical Physics},
year = {2019},
month = {May},
issn = {1664-2368},
keywords = {tracial positive linear map; Wigner--Yanase skew information; covariance; correlation; uncertainty relation},
}