Quantum information inequalities via tracial positive linear maps

TL;DR

This paper generalizes quantum information inequalities using tracial positive linear maps, establishing a noncommutative Heisenberg uncertainty relation and inequalities for generalized correlation and skew information.

Contribution

It introduces new quantum information inequalities involving tracial positive linear maps and extends the Heisenberg uncertainty principle to a noncommutative setting.

Findings

Established a noncommutative Heisenberg uncertainty relation.

Derived inequalities for generalized correlation and Wigner–Yanase–Dyson skew information.

Extended quantum information inequalities to the framework of $C^*$-algebras.

Abstract

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.