Journal of Mathematical Analysis and Applications, ( ISI ), Volume (447), No (1), Year (2017-3) , Pages (666-680)

#### Title : ( Quantum information inequalities via tracial positive linear maps )

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We present some generalizations of quantum information inequalities involving tracial positive linear maps between $C^*$-algebras. Among several results, we establish a noncommutative Heisenberg uncertainty relation. More precisely, we show that if $\Phi: \mathcal{A} \to \mathcal{B}$ is a tracial positive linear map between $C^*$-algebras , $\rho \in \mathcal{A}$ is a $\Phi$-density element and $A,B$ are self-adjoint operators of $\mathcal{A}$ such that ${\rm sp}(\mbox{-i}\rho^\frac{1}{2}[A,B]\rho^\frac{1}{2}) \subseteq [m,M]$ for some scalers $0<m<M$, then under some conditions \begin{eqnarray}\label{inemain1} V_{\rho,\Phi}(A)\sharp V_{\rho,\Phi}(B)\geq \frac{1}{2\sqrt{K_{m,M}(\rho[A,B])}} \left|\Phi(\rho [A,B])\right|, \end{eqnarray} where $K_{m,M}(\rho[A,B])$ is the Kantorovich constant of the operator $\mbox{-i}\rho^\frac{1}{2}[A,B]\rho^\frac{1}{2}$ and $V_{\rho,\Phi}(X)$ is the generalized variance of $X$.\\ In addition, we use some arguments differing from the scalar theory to present some inequalities related to the generalized correlation and the generalized Wigner--Yanase--Dyson skew information.

#### Keywords

, Tracial positive linear map; Heisenberg uncertainty relation; generalized covariance, generalized variance; generalized correlation; generalized Wigner--Yanase skew information;
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@article{paperid:1061842,
title = {Quantum information inequalities via tracial positive linear maps},
journal = {Journal of Mathematical Analysis and Applications},
year = {2017},
volume = {447},
number = {1},
month = {March},
issn = {0022-247X},
pages = {666--680},
numpages = {14},
keywords = {Tracial positive linear map; Heisenberg uncertainty relation; generalized covariance; generalized variance; generalized correlation; generalized Wigner--Yanase skew information; $C^*$-algebra},
}