Quantum information inequalities via tracial positive linear maps

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.