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Chi-Kwong Li

Publications and source records attributed to Chi-Kwong Li.

At least 55 records · Page 3Linked to original sources
arXiv2020

Joint numerical ranges and communtativity of matrices

The connection between the commutativity of a family of $n\times n$ matrices and the generalized joint numerical ranges is studied. For instance, it is shown that ${\cal F}$ is a family of mutually commuting normal matrices if and only if the joint numerical range $W_k(A_1, \dots, A_m)$ is a polyhedral set for some $k$ satisfying $|n/2-k|\le 1$, where $\{A_1, \dots, A_m\}$ is a basis for the linear span of the family; equivalently, $W_k(X,Y)$ is polyhedral for any two $X, Y \in {\cal F}$. More generally, characterization is given for the $c$-numerical range $W_c(A_1, \dots, A_m)$ to be polyhedral for any $n\times n$ matrices $A_1, \dots, A_m$. Other results connecting the geometrical properties of the joint numerical ranges and the algebraic properties of the matrices are obtained. Implications of the results to representation theory, and quantum information science are discussed.

math.FA↗
arXiv2019

Eigenvalue continuity and Geršgorin's theorem

Two types of eigenvalue continuity are commonly used in the literature. However, their meanings and the conditions under which continuities are used are not always stated clearly. This can lead to some confusion and needs to be addressed. In this note, we revisit the Geršgorin disk theorem and clarify the issue concerning the proofs of the theorem by continuity.

math.SP↗
arXiv2019

Higher Rank Matricial Ranges and Hybrid Quantum Error Correction

We introduce and initiate the study of a family of higher rank matricial ranges, taking motivation from hybrid classical and quantum error correction coding theory and its operator algebra framework. In particular, for a noisy quantum channel, a hybrid quantum error correcting code exists if and only if a distinguished special case of the joint higher rank matricial range of the error operators of the channel is non-empty. We establish bounds on Hilbert space dimension in terms of properties of a tuple of operators that guarantee a matricial range is non-empty, and hence additionally guarantee the existence of hybrid codes for a given quantum channel. We also discuss when hybrid codes can have advantages over quantum codes and present a number of examples.

quant-ph↗
arXiv2019

Numerical Range Inclusion, Dilation, and Operator Systems

Researchers have identified complex matrices $A$ such that a bounded linear operator $B$ acting on a Hilbert space will admit a dilation of the form $A \otimes I$ whenever the numerical range inclusion relation $W(B) \subseteq W(A)$ holds. Such an operator $A$ and the identity matrix will span a maximal operator system, i.e., every unital positive map from ${\rm span} \{I, A, A^*\}$ to ${\cal B}({\cal H})$, the algebra of bounded linear operators acting on a Hilbert space ${\cal H}$, is completely positive. In this paper, we identify $m$-tuple of matrices ${\bf A} = (A_1, \dots, A_m)$ such that any $m$-tuple of operators ${\bf B} = (B_1, \dots, B_m)$ satisfying the joint numerical range inclusion $W({\bf B}) \subseteq {\rm conv} W({\bf A})$ will have a joint dilation of the form $(A_1\otimes I, \dots, A_m\otimes I)$. Consequently, every unital positive map from ${\rm span} \{I, A_1, A_1^*, \dots, A_m, A_m^*\}$ to ${\cal B}({\cal H})$ is completely positive. New results and techniques are obtained relating to the study of numerical range inclusion, dilation, and maximal operator systems.

math.FA↗
arXiv2019

Preservation of the joint essential matricial range

Let $A = (A_1, \dots, A_m)$ be an $m$-tuple of elements of a unital $C$*-algebra ${\cal A}$ and let $M_q$ denote the set of $q \times q$ complex matrices. The joint $q$-matricial range $W^q(A)$ is the set of $(B_1, \dots, B_m) \in M_q^m$ such that $B_j = Φ(A_j)$ for some unital completely positive linear map $Φ: {\cal A} \rightarrow M_q$. When ${\cal A}= B(H)$, where $B(H)$ is the algebra of bounded linear operators on the Hilbert space $H$, the {\bf joint spatial $q$-matricial range} $W^q_s(A)$ of $A$ is the set of $(B_1, \dots, B_m) \in M_q^m$ for which there is a $q$-dimensional $V$ of $H$ such that $B_j$ is a compression of $A_j$ to $V$ for $j=1,\dots, m$. Suppose $K(H)$ is the set of compact operators in $B(H)$. The joint essential spatial $q$-matricial range is defined as $$W_{ess}^q(A) = \cap \{ {\bf cl}(W_s^q(A_1+K_1, \dots, A_m+K_m)): K_1, \dots, K_m \in K(H) \},$$ where ${\bf cl}$ denotes the closure. Let $π$ be the canonical surjection from $B(H)$ to the Calkin algebra $B(H)/K(H)$. We prove that $W_{ess}^q(A) =W^q(π(A) $, where $π(A) = (π(A_1), \dots, π(A_m))$. Furthermore, for any positive integer $N$, we prove that there are self-adjoint compact operators $K_1, \dots, K_m$ such that $${\bf cl}(W^q_s(A_1+K_1, \dots, A_m+K_m)) = W^q_{ess}(A) \quad \hbox{ for all } q \in \{1, \dots, N\}.$$ These results generalize those of Narcowich-Ward and Smith-Ward, obtained in the $m=1$ case, and also generalize a result of Müller obtained in case $m \ge 1$ and $q=1$. Furthermore, if $W_{ess}^1({\bf A}) $ is a simplex in ${\mathbb R}^m$, then we prove that there are self-adjoint $K_1, \dots, K_m \in K(H)$ such that ${\bf cl}(W^q_s(A_1+K_1, \dots, A_m+K_m)) = W^q_{ess}(A)$ for all positive integers $q$.

math.FA↗
arXiv2019

Nonsurjective maps between rectangular matrix spaces preserving disjointness, triple products, or norms

Let $M_{m,n}$ be the space of $m\times n$ real or complex rectangular matrices. Two matrices $A, B \in M_{m,n}$ are disjoint if $A^*B = 0_n$ and $AB^* = 0_m$. In this paper, a characterization is given for linear maps $Φ: M_{m,n} \rightarrow M_{r,s}$ sending disjoint matrix pairs to disjoint matrix pairs, i.e., $A, B \in M_{m,n}$ are disjoint ensures that $Φ(A), Φ(B) \in M_{r,s}$ are disjoint. More precisely, it is shown that $Φ$ preserves disjointness if and only if $Φ$ is of the form $$Φ(A) = U\begin{pmatrix} A \otimes Q_1 & 0 & 0 \cr 0 & A^t \otimes Q_2 & 0 \cr 0 & 0 & 0 \cr\end{pmatrix}V$$ for some unitary matrices $U \in M_{r,r}$ and $V\in M_{s,s}$, and positive diagonal matrices $Q_1, Q_2$, where $Q_1$ or $Q_2$ may be vacuous. The result is used to characterize nonsurjective linear maps that preserve the $JB^*$-triple product, or just the zero triple product, on rectangular matrices, defined by $\{A,B,C\} = \frac{1}{2}(AB^*C+CB^*A)$. The result is also applied to characterize linear maps between rectangular matrix spaces of different sizes preserving the Schatten $p$-norms or the Ky Fan $k$-norms.

math.RA↗
arXiv2019

Numerical Range Inclusion, Dilation, and completely positive maps

A proof using the theory of completely positive maps is given to the fact that if $A \in M_2$, or $A \in M_3$ has a reducing eigenvalue, then every bounded linear operator $B$ with $W(B) \subseteq W(A)$ has a dilation of the form $I \otimes A$. This gives a unified treatment for the different cases of the result obtained by researchers using different techniques.

math.FA↗
arXiv2019

Mathematical aspect of the combinatorial game "Mahjong"

We illustrate how one can use basic combinatorial theory and computer programming technique (Python) to analyze the combinatorial game: Mahjong. The results confirm some folklore concerning the game, and expose some unexpected results. Related results and possible future research in connection to artificial intelligence are mentioned. Readers interested in the subject may further develop the techniques to deepen the study of the game, or study other combinatorial games.

math.HO↗
arXiv2018

Some notes on the robustness of k-coherence and k-entanglement

We show that two related measures of k-coherence, called the standard and generalized robustness of k-coherence, are equal to each other when restricted to pure states. As a direct application of the result, we establish an equivalence between two analogous measures of Schmidt rank k-entanglement for all pure states. This answers conjectures raised in the literature regarding the evaluation of the quantifiers, and facilitates an efficient quantification of pure-state resources by introducing computable closed-form expressions for the two measures.

quant-ph↗
arXiv2018

Ranks of quantum states with prescribed reduced states

Let $M_n$ be the set of $n\times n$ complex matrices. In this note we determine all the possible ranks of a bipartite state in $M_m\otimes M_n$ with prescribed reduced states in the two subsystems. The results are used to determine the Choi rank of quantum channels $Φ: M_m \rightarrow M_n$ sending $I/m$ to a specific state $σ_2 \in M_n$.

quant-ph↗
arXiv2018

Minimum number of non-zero-entries in a $7\times 7$ stable matrix

We prove that if a $7\times 7$ matrix is potentially stable, then it has at least 11 non-zero entries. The results for $n\times n$ matrix with $n$ up to 6 are known previously. We prove the result by making a list of possible associated digraphs with at most 10 edges, and then use algebraic conditions to show all of these digraphs or matrices cannot be potentially stable. In relation to this, we also determine the minimum number of edges in a strongly connected digraph depending on its circumference.

math.CO↗
arXiv2018

The generalized numerical range of a set of matrices

For a given set of $n\times n$ matrices $\mathcal F$, we study the union of the $C$-numerical ranges of the matrices in the set $\mathcal F$, denoted by $W_C({\mathcal F})$. We obtain basic algebraic and topological properties of $W_C({\mathcal F})$, and show that there are connections between the geometric properties of $W_C({\mathcal F})$ and the algebraic properties of $C$ and the matrices in ${\mathcal F}$. Furthermore, we consider the starshapedness and convexity of the set $W_C({\mathcal F})$. In particular, we show that if ${\mathcal F}$ is the convex hull of two matrices such that $W_C(A)$ and $W_C(B)$ are convex, then the set $W_C({\mathcal F})$ is star-shaped. We also investigate the extensions of the results to the joint $C$-numerical range of an $m$-tuple of matrices.

math.FA↗
arXiv2018

Convexity and Star-shapedness of Matricial Range

Let ${\bf A} = (A_1, \dots, A_m)$ be an $m$-tuple of bounded linear operators acting on a Hilbert space ${\cal H}$. Their joint $(p,q)$-matricial range $Λ_{p,q}({\bf A})$ is the collection of $(B_1, \dots, B_m) \in {\bf M}_q^m$, where $I_p\otimes B_j$ is a compression of $A_j$ on a $pq$-dimensional subspace. This definition covers various kinds of generalized numerical ranges for different values of $p,q,m$. In this paper, it is shown that $Λ_{p,q}({\bf A})$ is star-shaped if the dimension of $\cal H$ is sufficiently large. If $\dim {\cal H}$ is infinite, we extend the definition of $Λ_{p,q}({\bf A})$ to $Λ_{\infty,q}({\bf A})$ consisting of $(B_1, \dots, B_m) \in {\bf M}_q^m$ such that $I_\infty \otimes B_j$ is a compression of $A_j$ on a closed subspace of ${\cal H}$, and consider the joint essential $(p,q)$-matricial range $$Λ^{ess}_{p,q}({\bf A}) = \bigcap \{ {\bf cl}(Λ_{p,q}(A_1+F_1, \dots, A_m+F_m)): F_1, \dots, F_m \hbox{ are compact operators}\}.$$ Both sets are shown to be convex, and the latter one is always non-empty and compact.

math.FA↗
arXiv2017

The modified trace distance of coherence is constant on most pure states

Recently, the much-used trace distance of coherence was shown to not be a proper measure of coherence, so a modification of it was proposed. We derive an explicit formula for this modified trace distance of coherence on pure states. Our formula shows that, despite satisfying the axioms of proper coherence measures, it is likely not a good measure to use, since it is maximal (equal to 1) on all except for an exponentially-small (in the dimension of the space) fraction of pure states.

quant-ph↗
arXiv2017

Unitary similarity invariant function preservers of skew products of operators

Let ${\mathcal B}(H)$ denote the Banach algebra of all bounded linear operators on a complex Hilbert space $H$ with $\dim H\geq 3$, and let $\mathcal A$ and $\mathcal B$ be subsets of ${\mathcal B}(H)$ which contain all rank one operators. Suppose $F(\cdot )$ is a unitary invariant norm, the pseudo spectra, the pseudo spectral radius, the $C$-numerical range, or the $C$-numerical radius for some finite rank operator $C$. The structure is determined for surjective maps $Φ:{\mathcal A}\rightarrow \mathcal B$ satisfying $F(A^*B)=F(Φ(A)^*Φ(B))$ for all $A, B \in {\mathcal A}$. To establish the proofs, some general results are obtained for functions $F:{\mathcal F}_1(H) \cup \{0\} \rightarrow [0, +\infty)$, where ${\mathcal F}_1(H)$ is the set of rank one operators in ${\mathcal B}(H)$, satisfying (a) $F(μUAU^*)=F(A)$ for a complex unit $μ$, $A\in {\mathcal F}_1(H)$ and unitary $U \in {\mathcal B}(H)$ (b) for any rank one operator $X\in {\mathcal F}_1(H)$ the map $t\mapsto F(tX)$ on $[0, \infty)$ is strictly increasing, and (c) the set $\{F(X): X \in {\mathcal F}_1(H) \hbox{ and } \|X\| = 1\}$ attains its maximum and minimum.

math.FA↗
arXiv2016

Numerical Ranges of the product of Operators

We study containment regions of the numerical range of the product of operators $A$ and $B$ such that $W(A)$ and $W(B)$ are line segments. It is shown that the containment region is equal to the convex hull of elliptical disks determined by the spectrum of $AB$, and conditions on $A$ and $B$ for the set equality holding are obtained. The results cover the case when $A$ and $B$ are self-adjoint operators extending the previous results on the numerical range of the product of two orthogonal projections.

math.FA↗
arXiv2016

Inequalities on generalized matrix functions

We prove inequalities on non-integer powers of products of generalized matrices functions on the sum of positive semi-definite matrices. For example, for any real number $r \in \{1\} \cup [2, \infty)$, positive semi-definite matrices $A_i,\ B_i,\ C_i\in M_{n_i}$, $i=1,2$, and generalized matrix functions $d_χ, d_ξ$ such as the determinant and permanent, etc., we have \begin{eqnarray*}&&\left(d_χ(A_1+B_1+C_1)d_ξ(A_2+B_2+C_2)\right)^r \\ &&\hskip 1in + \left(d_χ(A_1)d_ξ(A_2)\right)^r + \left(d_χ(B_1)d_ξ(B_2)\right)^r + \left(d_χ(C_1)d_ξ(C_2)\right)^r \\ & \ge &\left(d_χ(A_1+B_1 )d_ξ(A_2+B_2 )\right)^r + \left(d_χ(A_1+ C_1)d_ξ(A_2+ C_2)\right)^r + \left(d_χ( B_1+C_1)d_ξ( B_2+C_2)\right)^r\,.\end{eqnarray*} A general scheme is introduced to prove more general inequalities involving $m$ positive semi-definite matrices for $m \ge 3$ that extend the results of other authors.

math.FA↗