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 $Φ: \mathcal{A} \to \mathcal{B}$ is a tracial positive linear map between $C^*$-algebras , $ρ\in \mathcal{A}$ is a $Φ$-density element and $A,B$ are self-adjoint operators of $\mathcal{A}$ such that $ {\rm sp}(\mbox{-i}ρ^\frac{1}{2}[A,B]ρ^\frac{1}{2}) \subseteq [m,M] $ for some scalers $0<m<M$, then under some conditions \begin{eqnarray}\label{inemain1} V_{ρ,Φ}(A)\sharp V_{ρ,Φ}(B)\geq \frac{1}{2\sqrt{K_{m,M}(ρ[A,B])}} \left|Φ(ρ[A,B])\right|, \end{eqnarray} where $K_{m,M}(ρ[A,B])$ is the Kantorovich constant of the operator $\mbox{-i}ρ^\frac{1}{2}[A,B]ρ^\frac{1}{2}$ and $V_{ρ,Φ}(X)$ 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.