arXiv subjects

Ali Dadkhah

Publications and source records attributed to Ali Dadkhah.

3 recordsLinked to original sources

arXiv2019

Noncommutative versions of inequalities in quantum information theory

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 $x^α$ and $x^{1-α}$ 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 persent 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.

math.OA↗

arXiv2018

Non-linear positive maps between $C^*$-algebras

We present some properties of (not necessarily linear) positive maps between $C^*$-algebras. We first extend the notion of Lieb functions to that of Lieb positive maps between $C^*$-algebras. Then we give some basic properties and fundamental inequalities related to such maps. Next, we study $n$-positive maps ($n\geq 2$). We show that if for a unital $3$-positive map $Φ: \mathscr{A}\longrightarrow\mathscr{B}$ between unital $C^*$-algebras and some $ A\in \mathscr{A}$ equality $Φ(A^*A)= Φ(A)^* Φ(A)$ holds, then $Φ(XA)=Φ(X)Φ(A)$ for all $X \in \mathscr{A}$. In addition, we prove that for a certain class of unital positive maps $Φ: \mathscr{A}\longrightarrow\mathscr{B}$ between unital $C^*$-algebras, the inequality $Φ(αA)\leqαΦ(A)$ holds for all $ α\in [0,1]$ and all positive elements $ A\in \mathscr{A}$ if and only if $Φ(0)=0$. Furthermore, we show that if for some $α$ in the unit ball of $\mathbb{C}$ or in $\mathbb{R}_+$ with $|α|\neq 0,1$, the equality $Φ(αI)=αI$ holds, then $Φ$ is additive on positive elements of $\mathscr{A}$. Moreover, we present a mild condition for a $6$-positive map, which ensures its linearity.

math.OA↗

arXiv2016

Grüss type inequalities for positive linear maps on $C^*$-algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be two unital $C^*$-algebras and let for $C\in\mathcal{A},\ Γ_C=\{γ\in \mathbb{C} : \|C-γI\|=\inf_{α\in \mathbb{C}} \|C-αI\|\}$. We prove that if $Φ:\mathcal{A} \longrightarrow \mathcal{B}$ is a unital positive linear map, then \begin{eqnarray*} \big|Φ(AB)-Φ(A)Φ(B)\big| \leq \big\|Φ(|A^*-ζI|^2)\big\|^\frac{1}{2} \big[Φ(|B-ξI|^2)\big]^\frac{1}{2} \end{eqnarray*} for all $A,B\in\mathcal{A}, ζ\in Γ_A$ and $ξ\inΓ_B.$\\ In addition, we show that if $(\mathcal{A},τ)$ is a noncommutative probability space and $T \in \mathcal{A}$ is a density operator, then \begin{eqnarray*} \ \ \big|τ(TAB)-τ(TA)τ(TB)\big|\leq \|A-ζI\|_p\|B-ξI\|_q\|T\|_r \ \ (p,q\geq 4, r\geq 2) \end{eqnarray*} and \begin{eqnarray*} \big|τ(TAB)-τ(TA)τ(TB)\big|\leq \|A-ζI\|_p\|B-ξI\|_q\|T\| \ \ \ \ (p,q\geq 2)\ \ \ \ \ \end{eqnarray*} for every $A,B \in \mathcal{A}$ and $ζ\in Γ_A,ξ\in Γ_B$. Our results generalize the corresponding results for matrices to operators on spaces of arbitrary dimension.

math.OA↗