In a recent work, Moslehian and Rajic have shown that the Gruss inequality holds for unital n-positive linear maps $ϕ:\mathcal A \rightarrow B(H)$, where $\mathcal A$ is a unital C*-algebra and H is a Hilbert space, if $n \ge 3$. They also demonstrate that the inequality fails to hold, in general, if $n = 1$ and question whether the inequality holds if $n=2$. In this article, we provide an affirmative answer to this question.
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{e…
Assuming a unitarily invariant norm $|||\cdot|||$ is given on a two-sided ideal of bounded linear operators acting on a separable Hilbert space, it induces some unitarily invariant norms $|||\cdot|||$ on matrix algebras $\mathcal{M}_n$ for all finite values of $n$ via $|||A|||=|||A\oplus 0|||$. We show that if $\mathscr{A}$ is a $C^*$-algebra of finite dimension $k$ and $Φ: \mathscr{A} \to \mathcal{M}_n$ is a unital completely positive map, then \begin{equation*} |||Φ(AB)-Φ(A)Φ(B)||| \leq \frac{1}{4} |||I_{n}|||\,|||I_{kn}||| d_A d_B \end{equation*} for any $A,B \in \mathscr{A}$, where $d_X$ d…
Bounds on the distance between a unital quantum channel and the convex hull of unitary channels, with applications to the asymptotic quantum Birkhoff conjecture
We prove some refinements of a reverse AM-GM operator inequality due to M. Lin [Studia Math. 2013;215:187-194]. In particular, we show the operator inequality \begin{eqnarray*} Φ^p\left(A\nabla_νB+2rMm(A^{-1}\nabla B^{-1}-A^{-1}\sharp B^{-1})\right)\leqα^pΦ^p\left(A\sharp_νB\right), \end{eqnarray*} where $A,B$ are positive operators on a Hilbert space such that $0<m \leq A, B \leq M$ for some positive numbers $m, M$, $Φ$ is a positive unital linear map, $ν\in[0,1]$, $r=\min\{ν,1-ν\}$, $p>0$ and $α=\max\left\{\frac{(M+m)^2}{4Mm},\frac{(M+m)^2}{4^\frac{2}{p}Mm}\right\}$.
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 certa…
We define the graph product of unital completely positive maps on a universal graph product of unital C*-algebras and show that it is unital completely positive itself. To accomplish this, we introduce the notion of the non-commutative length of a word, and we obtain a Stinespring construction for concatenation. This result yields the following consequences. The graph product of positive-definite functions is positive-definite. A graph product version of von Neumann's Inequality holds. Graph independent contractions on a Hilbert space simultaneously dilate to graph independent unitaries.
We study quantum computation relations on unital finite-dimensional CAR $C^{*}$-algebras. We prove an entropy power inequality (EPI) in a fermionic setting, which presumably will permit understanding the capacities in fermionic linear optics. Similar relations to the bosonic case are shown, and alternative proofs of known facts are given. Clifford algebras and the Grassmann representation can thus be used to obtain mathematical results regarding coherent fermion states.
Let $\mathscr{R}$ be a finite von Neumann algebra with a faithful tracial state $τ$ and let $Δ$ denote the associated Fuglede-Kadison determinant. In this paper, we characterize all unital bijective maps $ϕ$ on the set of invertible positive elements in $\mathscr{R}$ which satisfy $$Δ(ϕ(A)+ϕ(B)) = Δ(A+B).$$ We show that any such map originates from a $τ$-preserving Jordan $*$-automorphism of $\mathscr{R}$ (either $*$-automorphism or $*$-anti-automorphism in the more restrictive case of finite factors). In establishing the aforementioned result, we make crucial use of the solutions to the equat…
Let $\mathcal{A}$ be a unital C*-algebra. Then the theory of Hilbert C*-modules tells that \begin{align*} \sum_{i=1}^{n}(a_ia_i^*)^\frac{1}{2}\leq \sqrt{n} \left(\sum_{i=1}^{n}a_ia_i^*\right)^\frac{1}{2}, \quad \forall n \in \mathbb{N}, \forall a_1, \dots, a_n \in \mathcal{A}. \end{align*} By modifications of arguments of Botelho-Andrade, Casazza, Cheng, and Tran given in 2019, for certain tuple $x=(a_1, \dots, a_n) \in \mathcal{A}^n$, we give a method to compute a positive element $c_x$ in the C*-algebra $\mathcal{A}$ such that the equality \begin{align*} \sum_{i=1}^{n}(a_ia_i^*)^\frac{1}{2}=…
Sufficient positive maps between von Neumann algebras: Rényi divergences
Mathematical Institute of the Slovak Academy of Sciences（斯洛伐克科学院数学研究所）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）；Institute for Quantum Computing（量子计算研究所）
