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Chi-Kwong Li

Publications and source records attributed to Chi-Kwong Li.

At least 19 recordsLinked to original sources
arXiv2026

Geometry of essential matrix ranges and the Smith-Ward problem for operator systems

A $d$-tuple of bounded linear selfadjoint operators acting on an infinite-dimensional separable Hilbert space is said to have the Smith-Ward property if the identity map of the image of the operator system in the Calkin algebra has a completely positive lift. In this paper, we focus on noncommutative geometric properties of a finite dimensional operator system with the goal of understanding how geometric information encoded by the essential matrix range of a spanning set of linear basis for the operator system implies the Smith-Ward property. Some geometric objects of special interest in this paper include one form of noncommutative complex Euclidean ball and maximal noncommutative cubes and polydiscs, as well as some extremal compact matrix convex sets, $K^{\rm min}$ or $K^{\rm max}$, determined by a given compact convex subset $K$ of $\mathbb R^d$.

math.OA↗
arXiv2026

Numerical radius of certain two-by-two block matrices

We investigate the numerical range $W(T)$ and numerical radius $w(T)$ of operators of the form $T = \begin{pmatrix} A & B \\ 0 & 0 \end{pmatrix}$. We show that $W(T)$ is the union of the numerical ranges of a family of $2\times 2$ matrices, $T_x$, leading to several consequences, including improved inequalities for $w(T)$. For cases where $A$ is a self-adjoint involution, we characterize the conditions under which $W(T)$ is an elliptical disk and determine the minimum numerical radius of $T_U = \begin{pmatrix} U^*AU & B \\ 0 & 0 \end{pmatrix}$ over all unitary operators $U$. Finally, we study matrices $T \in M_n$ satisfying $\|T^m x\| = \|T^m\| = \|T\|$ for a unit vector $x$ and all positive integers $m$. This analysis connects these matrices to the aforementioned block form and provides a counterexample to the conjecture that if $\|T^k\| = \|T\|$ for all $k \ge 1$, then some power of the matrix has a direct summand that is a scalar multiple of an idempotent.

math.FA↗
arXiv2026

Preserver problems on Toeplitz matrices

\We study linear preserver problems on the linear space of $n\times n$ Toeplitz matrices over the real field or the complex field. In particular, characterizations are given for linear preservers of rank one matrices and linear preservers of the determinant. We also present related results and questions on other structured matrices.

math.FA↗
arXiv2026

Generalized matrix nearness problems II

Given a matrix $A$, a matrix nearness problem seeks an $X$ that most closely approximates $A$ in the sense of minimizing $\lVert A - X\rVert$ under a variety of constraints on $X$. A generalized matrix nearness problem seeks the same but with three given matrices $A,B,C$ and $\lVert A - BXC\rVert$ in place of $\lVert A - X\rVert$. We extend previous studies of the latter problem in three directions: incorporating an affine term, replacing matrix product by Kronecker product in various manners, and generalizing Frobenius norm to any orthogonally invariant norm. We will solve several of these in closed form. For the rest, we develop an iterative algorithm that works for any Schatten norm, proving that it converges to a global minimizer regardless of the initial point. In addition, the algorithm relies purely on numerical linear algebra, and notably does not compute any explicit gradients or subgradients. Along the way, we will also show that there is no Mirsky-type theorem for rank constrained generalized matrix nearness problems.

math.NA↗
arXiv2026

Linear maps preserving product of involutions

An element of the algebra $M_n(\mathbb{F})$ of $n \times n$ matrices over a field $\mathbb{F}$ is called an involution if its square equals the identity matrix. Gustafson, Halmos, and Radjavi proved that any product of involutions in $M_n(\mathbb{F})$ can be expressed as a product of at most four involutions. In this article, we investigate the bijective linear preservers of the sets of products of two, three, or four involutions in $M_n(\mathbb{F})$.

math.FA↗
arXiv2026

Approximating quantum states by states of low rank

Given a positive integer k, it is natural to ask for a formula for the distance between a given density matrix (i.e., mixed quantum state) and the set of density matrices of rank at most k. This problem has already been solved when "distance" is measured in the trace or Frobenius norm. We solve it for all other unitary similarity invariant norms. We also present some consequences of our formula. For example, in the trace and Frobenius norms, the density matrix that is farthest from the set of low-rank density matrices is the maximally-mixed state, but this is not true in many other unitary similarity invariant norms.

quant-ph↗
arXiv2026

Semigroup automorphisms of total positivity

Totally positive (TP) and totally nonnegative (TN) matrices connect to analysis, mechanics, and to dual canonical bases in reductive groups, by well-known works of Schoenberg, Gantmacher-Krein, Lusztig, and others. TP matrices form a multiplicatively closed semigroup, contained in the larger monoid of invertible totally nonnegative (ITN) matrices. Whitney and Berenstein-Fomin-Zelevinsky found bidiagonal factorizations of all $n\times n$ ITN and TP matrices into multiplicative generators; a natural question now is to classify the multiplicative automorphisms of these semigroups. In this article, we classify all automorphisms of these semigroups of ITN and TP matrices. In particular, we show that the automorphisms are the same, and they respect the multiplicative generators.

math.RA↗
arXiv2026

Cliques and independent subgroups of the Birkhoff polytope graph

The Birkhoff polytope $Ω_n$ is the polytope of doubly stochastic matrices of order $n$. The Birkhoff polytope graph $G(Ω_n)$ is the skeleton of $Ω_n$; it is the Cayley graph whose vertex set consists of the elements of the symmetric group ${\rm Sym}(n)$ of degree $n$, where two permutations are adjacent if one equals the product of the other with a cycle. We study the combinatorial structure of this graph, focusing on its maximal and maximum cliques and on its independent subgroups (subgroups of ${\rm Sym}(n)$ whose elements are pairwise nonadjacent in the graph). We obtain maximal subgroups of $G(Ω_n)$ and establish both a lower bound and an upper bound for its clique number. Especially, we prove that if $K$ is a subset of ${\rm Sym}(n)$ consisting of 3-cycle permutations such that $δ_1^{-1}δ_2$ is a single cycle for all $δ_1,δ_2\in K$, then the maximum size of $K$ is $\lfloor (n-1)^2/4\rfloor$, which can be viewed as an Erdős-Ko-Rado-type theorem for ${\rm Sym}(n)$.

math.CO↗
arXiv2025

Efficient Circuit-Based Quantum State Tomography via Sparse Entry Optimization

Many quantum states arising in algorithms and physical systems occupy only a small, structured subset of the exponentially large Hilbert space, yet standard quantum state tomography fails to exploit this structure. We present an efficient circuit-based tomography framework for pure quantum states that are sparse in a computational basis. For an $n$-qubit state supported on $k$ basis elements, the protocol reconstructs all amplitudes using $1 + 2(k-1)$ measurement settings. The method admits both entanglement-assisted and entanglement-free implementations, enabling explicit tradeoffs between two-qubit gate usage and statistical noise. We derive bounds on the required number of CNOT gates from the combinatorial structure of the state support and analyze their effect on reconstruction infidelity. The framework extends naturally to closed-system process tomography and is validated via numerical simulations using Qiskit.

quant-ph↗
arXiv2025

An edge-based and subspace reduction encoding scheme to solve the traveling salesman problem in quantum computers

This paper introduces a novel edge-based encoding technique for solving the Traveling Salesman Problem (TSP) on a quantum computer, reducing the required number of qubits. For implementation in real quantum devices, we applied the subspace reduction encoding to further reduce the dimension of the TSP solution space. We attack the TSP for 4-, 5-, and 6-city instances in both simulators and real quantum computers across different encoding frameworks. Optimal solutions of the 4-city TSP instance are obtained on state-of-the art IQM quantum computer. Our study presents a comparative analysis between edge-based encoding scheme and the node-based encoding methodology in the literature. Our findings indicate that the proposed encoding scheme outperforms conventional methods in terms of statistical measures, quantum resource utilization, and computational efficiency when applied to smaller TSP instances.

quant-ph↗
arXiv2025

(Real)linear preservers of multiples of unitaries and matrix pairs with some extremal norm properties

We determine the structure of linear maps on complex (real) square matrices sending unitary (orthogonal) matrices to multiples of unitary (orthogonal) matrices. The result is used to determine the linear preservers of matrix pairs satisfying the extremal norm properties $\|AB\| = \|A\| \|B\|$, $\|A^*B\| = \|A\| \|B\|$, or $\|AB^*\| = \|A\| \|B\|$, for the spectral norm $\|\cdot\|$.

math.FA↗
arXiv2025

Confluent Vandermonde matrix and related topics

In this note, we explore the connections between the confluent Vandermonde matrix over an arbitrary field and several mathematical topics, including interpolation polynomials, Hasse derivatives, LU factorization, companion matrices and their Jordan forms, and the partial fraction decomposition. Using a unified approach based on polynomial evaluations and derivative computations at selected points, we provide accessible proofs that not only clarify key results but also offer insights for both experienced researchers and those new to the subject.

math.CO↗
arXiv2025

A Note on Eigenvalues of Perturbed Hermitian Matrices

Let $$ A=\left(\begin{array}{cc} H_1 & E^*\\ E & H_2\end{array}\right) \quad \hbox{ and } \quad \wtd A=\left(\begin{array}{cc} H_1 & O\\ O & H_2\end{array}\right)$$ be two $N$-by-$N$ Hermitian matrices with eigenvalues $λ_1 \ge \cdots \ge λ_{N}$ and $\wtd λ_1 \ge \cdots \ge \wtd λ_N$, respectively. \iffalse There are two kinds of perturbation bounds on $|λ_i - \wtd λ_i|$: $|λ_i- \wtd λ_i| \le \|E\|$, where $\|E\|$ is the largest singular value of $\|E\|$, regardless of $H_i$'s spectral distributions, and $|λ_i - \wtd λ_i| \le \|E\|^2/η$, where $η$ is the minimum gap between $H_i$'s spectra. \end{enumerate} Bounds of the first kind overestimate the changes when $\|E\|\llη$ while those of the second kind may blow up when $η$ is too tiny. \fi Denote by $\|E\|$ the spectral norm of the matrix $E$, and $η$ the spectral gap between the spectra of $H_1$ and $H_2$. It is shown that $$ |λ_i - \wtd λ_i| \le {2\|E\|^2 \over η+\sqrt{η^2+4\|E\|^2}} \, , $$ which improves all the existing results. Similar bounds are obtained for singular values of matrices under block perturbations.

math.NA↗
arXiv2024

Norm of an operator with numerical range in a sector

We refine a recent result of Drury concerning the optimal ratio between the norm and numerical radius of a bounded linear operator $T$ with numerical range lying in a sector of a circular disk. In particular, characterization is given to the operators attaining the optimal ratio, and properties of such operators are explored.

math.FA↗
arXiv2024

Linear preservers of parallel matrix pairs with respect to the $k$-numerical radius

Let $1 \leq k < n$ be integers. Two $n \times n$ matrices $A$ and $B$ form a parallel pair with respect to the $k$-numerical radius $w_k$ if $w_k(A + μB) = w_k(A) + w_k(B)$ for some scalar $μ$ with $|μ| = 1$; they form a TEA (triangle equality attaining) pair if the preceding equation holds for $μ= 1$. We classify linear bijections on $\mathbb M_n$ and on $\mathbb H_n$ which preserve parallel pairs or TEA pairs. Such preservers are scalar multiples of $w_k$-isometries, except for some exceptional maps on $\mathbb H_n$ when $n=2k$.

math.FA↗
arXiv2024

Linear maps preserving $\ell_p$-norm parallel vectors

Two vectors $x, y$ in a normed vector space are parallel if there is a scalar $μ$ with $|μ| = 1$ such that $\|x+μy\| = \|x\| + \|y\|$; they form a triangle equality attaining (TEA) pair if $\|x+y\| = \|x\| + \|y\|$. In this paper, we characterize linear maps on $F^n=R^n$ or $C^n$, equipped with the $\ell_p$-norm for $p \in [1, \infty]$, preserving parallel pairs or preserving TEA pairs. Indeed, any linear map will preserve parallel pairs and TEA pairs when $1< p <\infty$. For the $\ell_1$-norm, TEA preservers form a semigroup of matrices in which each row has at most one nonzero entries; adding rank one matrices to this semigroup will be the semigroup of parallel preserves. For the $\ell_\infty$-norm, a nonzero TEA preserver, or a parallel preserver of rank greater than one, is always a multiple of an $\ell_\infty$-norm isometry, except when $F^n = R^2$. We also have a characterization for the exceptional case. The results are extended to linear maps of the infinite dimensional spaces $\ell_1(Λ)$, $c_0(Λ)$ and $\ell_\infty(Λ)$.

math.FA↗
arXiv2024

Linear maps on matrices preserving parallel pairs

Two (real or complex) $m\times n$ matrices $A$ and $B$ are said to be parallel (resp. triangle equality attaining, or TEA in short) with respect to the spectral norm $\|\cdot\|$ if $\|A+ μB\| = \|A\| + \|B\|$ for some scalar $μ$ with $|μ|=1$ (resp. $μ=1$). We study linear maps $T$ on $m\times n$ matrices preserving parallel (resp. TEA) pairs, i.e., $T(A)$ and $T(B)$ are parallel (resp. TEA) whenever $A$ and $B$ are parallel (resp. TEA). It is shown that when $m,n \ge 2$ and $(m,n) \ne (2,2)$, a nonzero linear map $T$ preserving TEA pairs if and only if it is a positive multiple of a linear isometry, namely, $T$ has the form $$(1) \quad A \mapsto γUAV \quad \quad \text{or} \quad \quad (2) \quad A \mapsto γUA^{t} V \quad (\text{in this case}, m = n),$$ for a positive number $γ$, and unitary (or real orthogonal) matrices $U$ and $V$ of appropriate sizes. Linear maps preserving parallel pairs are those carrying form (1), (2), or the form $$ (3) \ A \mapsto f(A) Z$$ for a linear functional $f$ and a fixed matrix $Z$. The case when $(m,n) = (2,2)$ is more complicated. There are linear maps of $2\times 2$ matrices preserving parallel pairs or TEA pairs neither of the form (1), (2) nor (3) above. Complete characterization of such maps is given with some intricate computation and techniques in matrix groups.

math.RA↗