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Hao-Wei Huang

Publications and source records attributed to Hao-Wei Huang.

15 recordsLinked to original sources
arXiv2026

Scaling limit theorem for mixed free and Boolean convolution powers

We prove a scaling limit theorem for a double sequence of probability measures involving additive free convolution $\boxplus$ and additive Boolean convolution $\uplus$. Let $μ$ be a probability measure on $\mathbb{R}$ with mean zero and variance one, and let $M=M(N)>0$ satisfy $MN^{α+1/2}\to t>0$. We study the weak limits, as $N\to \infty$, of the double arrays $D_{N^α}((μ^{\boxplus N})^{\uplus M})$. We show that the limit distribution is the Cauchy distribution with scale parameter $t$ if $α>-1/2$, the $t$-fold Boolean convolution power of the standard semicircle law if $α=-1/2$, and the point mass at the origin if $α<-1/2$.

math.PR↗
arXiv2025

Sharp estimates of quantum covering problems via a novel trace inequality

In this paper, we prove a novel trace inequality involving two operators. As applications, we sharpen the one-shot achievability bound on the relative entropy error in a wealth of quantum covering-type problems, such as soft covering, privacy amplification, convex splitting, quantum information decoupling, and quantum channel simulation by removing some dimension-dependent factors. Moreover, the established one-shot bounds extend to infinite-dimensional separable Hilbert spaces as well. The proof techniques are based on the recently developed operator layer cake theorem and an operator change-of-variable argument, which are of independent interest.

quant-ph↗
arXiv2025

An Iterative Methodology for Unitary Quantum Channel Search

In this paper, we propose an iterative algorithm using polar decomposition to approximate a channel characterized by a single unitary matrix based on input-output quantum state pairs. In limited data, we state and prove that the optimal solution obtained from our method using one pair with a specific structure will generate an equivalent class, significantly reducing the dimension of the searching space. Furthermore, we prove that the unitary matrices describing the same channel differ by a complex number with modulus 1. We rigorously prove our proposed algorithm can ultimately identify a critical point, which is also a local minimum of the established objective function.

math.NA↗
arXiv2022

Regularity results for free Lévy processes

Given a free additive convolution semigroup $\left(μ_t\right)_{t\geq 0}$ and a probability measure $ν$ on $\mathbb{R}$, we find the necessary and sufficient conditions for the process $μ_t \boxplus ν$ to be Lebesgue absolutely continuous with a positive and analytic density throughout $\mathbb{R}$ at all time $t>0$. For semigroups without this property, we find the necessary and sufficient conditions for the density of $μ_t \boxplus ν$ to be analytic at its zeros. These results are quantified by the Lévy measure of the semigroup, making it fairly easy to construct many concrete examples. Finally, we show that $μ_t \boxplus ν$ has a finite number of connected components in its support if both the Lévy measure of $\left(μ_t\right)_{t \geq 0}$ and the initial law $ν$ do.

math.PR↗
arXiv2020

Limit Theorems and Wrapping Transforms in Bi-free Probability Theory

In this paper, we characterize idempotent distributions with respect to the bi-free multiplicative convolution on the bi-torus. Also, the bi-free analogous Levy triplet of an infinitely divisible distribution on the bi-torus without non-trivial idempotent factors is obtained. This triplet is unique and generates a homomorphism from the bi-free multiplicative semigroup of infinitely divisible distributions to the classical one. The relevances of the limit theorems associated with four convolutions, classical and bi-free additive convolutions and classical and bi-free multiplicative convolutions, are analyzed. The analysis relies on the convergence criteria for limit theorems and the use of push-forward measures induced by the wrapping map from the plane to the bi-torus. Different from the bi-free circumstance, the classical multiplicative Lévy triplet is not always unique. Due to this, some conditions are furnished to ensure uniqueness.

math.FA↗
arXiv2018

Bi-Free Extreme Values

In this paper, we continue Voiculescu's recent work on the analogous extreme value theory in the context of bi-free probability theory. We derive various equivalent conditions for a bivariate distribution function to be bi-freely max-infinitely divisible. A bi-freely max-infinitely divisible distribution function can be expressed in terms of its marginals and a special form of copulas. Such a distribution function is shown to be also max-infinitely divisible in the classical sense. In addition, we characterize the set of bi-free extreme value distribution functions. A distribution function of this type is also bi-free max-stable and represented by its marginals and one copula composing of a Pickands dependence function, as in the classical extreme value theory. As a consequence, the determination of its bi-free domain of attraction is the same as the criteria in the classical theory. To illustrate these connections, some concrete examples are provided.

math.OA↗
arXiv2017

Limit theorems in bi-free probability theory

In this paper additive bi-free convolution is defined for general Borel probability measures, and the limiting distributions for sums of bi-free pairs of selfadjoint commuting random variables in an infinitesimal triangular array are determined. These distributions are characterized by their bi-freely infinite divisibility, and moreover, a transfer principle is established for limit theorems in classical probability theory and Voiculescu's bi-free probability theory. Complete descriptions of bi-free stability and fullness of planar probability distributions are also set down. All these results reveal one important feature about the theory of bi-free probability that it parallels the classical theory perfectly well. The emphasis in the whole work is not on the tool of bi-free combinatorics but only on the analytic machinery.

math.PR↗
arXiv2017

Harmonic analysis for the bi-free partial S-transform

This is a continuation of our previous work in bi-free harmonic analysis for commuting left and right variables. Here we analyze the bi-free partial S-transform and use the results to study limit theorems and infinite divisibility relative to the multiplicative bi-free convolution.

math.OA↗
arXiv2016

Analytic aspects of the bi-free partial R-transform

Since Voiculescu introduced his bi-free probability theory in 2013, the major development of the theory has been on its combinatorial side; in particular, on the combinatorics of bi-free cumulants and its application to the bi-free R-transform. In this article we propose a harmonic analysis approach to the bi-free R-transform, which is solely based on integral transforms of two variables. To accommodate the harmonic analysis tools, we confine ourselves in the simplest situation of bi-freeness with commuting faces. Our method allows us to treat measures with unbounded support, and we show that the classical limit theory of infinitely divisible laws, due to Levy and Khintchine, has a perfect bi-free analogue.

math.OA↗
arXiv2015

An analogue of the Lévy-Hinčin formula for bi-free infinitely divisible distributions

In this paper, we derive the bi-free analogue of the Lévy-Hinčin formula for compactly supported planar probability measures which are infinitely divisible with respect to the additive bi-free convolution introduced by Voiculescu. We also provide examples of bi-free infinitely divisible distributions with their bi-free Lévy-Hinčin representations. Furthermore, we construct the bi-free Lévy processes and the additive bi-free convolution semigroups generated by compactly supported planar probability measures.

math.OA↗
arXiv2013

On the support of measures in multiplicative free convolution semigroups

In this paper, we study the supports of measures in multiplicative free semigroups on the positive real line and on the unit circle. We provide formulas for the density of the absolutely continuous parts of measures in these semigroups. The descriptions rely on the characterizations of the images of the upper half-plane and the unit disc under certain subordination functions. These subordination functions are $η$-transforms of infinitely divisible measures with respect to multiplicative free convolution. The characterizations also help us study the regularity properties of these measures. One of the main results is that the number of components in the support of measures in the semigroups is a decreasing function of the semigroup parameter.

math.CV↗
arXiv2012

Supports, regularity, and $\boxplus$-infinite divisibility for measures of the form $(μ^{\boxplus p})^{\uplus q}$

Let $\mathcal{M}$ be the set of Borel probability measures on $\mathbb{R}$. We denote by $μ^{\mathrm{ac}}$ the absolutely continuous part of $μ\in\mathcal{M}$. The purpose of this paper is to investigate the supports and regularity for measures of the form $(μ^{\boxplus p})^{\uplus q}$, $μ\in\mathcal{M}$, where $\boxplus$ and $\uplus$ are the operations of free additive and Boolean convolution on $\mathcal{M}$, respectively, and $p\geq1$, $q>0$. We show that for any $q$ the supports of $((μ^{\boxplus p})^{\uplus q})^{\mathrm{ac}}$ and $(μ^{\boxplus p})^{\mathrm{ac}}$ contain the same number of components and this number is a decreasing function of $p$. Explicit formulas for the densities of $((μ^{\boxplus p})^{\uplus q})^{\mathrm{ac}}$ and criteria for determining the atoms of $(μ^{\boxplus p})^{\uplus q}$ are given. Based on the subordination functions of free convolution powers, we give another point of view to analyze the set of $\boxplus$-infinitely divisible measures and provide explicit expressions for their Voiculescu transforms in terms of free and Boolean convolutions.

math.CV↗
arXiv2012

Supports of Measures in a free additive convolution semigroup

In this paper, we study the supports of measures in the free additive convolution semigroup $\{μ^{\boxplus t}:t>1\}$, where $μ$ is a Borel probability measure on $\mathbb{R}$. We give a formula for the density of the absolutely continuous part of $μ^{\boxplus t}$ and use this formula to obtain certain regularizing properties of $μ^{\boxplus t}$. We show that the number $n(t)$ of the components in the support of $μ^{\boxplus t}$ is a decreasing function of $t$ and give equivalent conditions so that $n(t)=1$ for sufficiently large $t$. Moreover, a measure $μ$ so that $μ^{\boxplus t}$ has infinitely many components in the support for all $t>1$ is given.

math.CV↗
arXiv2011

Closability property of operator algebras generated by normal operators and operators of class $C_0$

An operator algebra $\mathcal{A}$ acting on a Hilbert space is said to have the closability property if every densely defined linear transformation commuting with $\mathcal{A}$ is closable. In this paper we study the closability property of the von Neumann algebra consisting of the multiplication operators on $L^2(μ)$, and give necessary and sufficient conditions for a normal operator $N$ such that the von Neumann algebra generated by $N$ has the closability property. We also give necessary and sufficient conditions for an operator $T$ of class $C_0$ such that the algebra generated by $T$ in the weak operator topology and the algebra $H^\infty(T)=\{u(T):u\in H^\infty\}$ have the closability property.

math.OA↗