Diagonal Compressed Random Variables in a Graph W*-probability Spaces

Compressed Random Variables in a Graph W*-Probability Space

In this paper, we will consider the free probabilistic information about compressed random variables in a graph W*-Probability space. Recall the diagonal compressed random variables in a graph W*-probability space. In particular, we can see that the free moments and cumulants of the fixed compressed random variable of a random variable x are exactly same as the diagonal compressed part of x.

Eta-diagonal distributions and infinite divisibility for R-diagonals

The class of R-diagonal *-distributions is fairly well understood in free probability. In this class, we consider the concept of infinite divisibility with respect to the operation $\boxplus$ of free additive convolution. We exploit the relation between free probability and the parallel (and simpler) world of Boolean probability. It is natural to introduce the concept of an eta-diagonal distribution that is the Boolean counterpart of an R-diagonal distribution. We establish a number of properties of eta-diagonal distributions, then we examine the canonical bijection relating eta-diagonal distr…

Random Variables in Graph W*-Probability Spaces

In [16], we observed the graph W*-probability theory. In this paper, we will review [16] and introduce special amalgamated random variables in this amalgamated W*-probability space. In particular, we will observe the amalgamated semicircularity, amalgamated evenness and amalgamated R-diagonality. As an example, we will compute the trivial moments and trivial cumulants of the generating operator of the graph W*-algebra.

关于双自由概率中双R-对角算子对的性质

On Bi-R-Diagonal Pairs of Operators

研究双自由概率中R-对角算子的性质，分析双R-对角算子对的乘积及幂的性质，探讨其联合分布的不变性及等价刻画。

Density Matrix Diagonal-Block Lovas-Andai-type singular-value ratios for qubit-qudit separability/PPT probability analyses

An important variable in the 2017 analysis of Lovas and Andai, formally establishing the Hilbert-Schmidt separability probability conjectured by Slater of $\frac{29}{64}$ for the 9-dimensional convex set of two-rebit density matrices, was the ratio ($\varepsilon =\frac{σ_2}{σ_1}$) of the two singular values ($σ_1 \geq σ_2 \geq 0$) of $D_2^{\frac{1}{2}} D_1^{-\frac{1}{2}}$. There, $D_1$ and $D_2$ were the diagonal $2 \times 2$ blocks of a $4 \times 4$ two-rebit density matrix $ρ$. Working within the Lovas-Andai "separability function" ($\tildeχ_d(\varepsilon)$) framework, Slater was able to ver…

$R$-diagonal and $η$-diagonal Pairs of Random Variables

This paper is devoted to studying $R$-diagonal and $η$-diagonal pairs of random variables. We generalize circular elements to the bi-free setting, defining bi-circular element pairs of random variables, which provide examples of $R$-diagonal pairs of random variables. Formulae are given for calculating the distributions of the product pairs of two $*$-bi-free $R$-diagonal pairs. When focusing on pairs of left acting operators and right acting operators from finite von Neumann algebras in the standard form, we characterize $R$-diagonal pairs in terms of the $*$-moments of the random variables, …