arXiv subjects

Chi-Kwong Li

Publications and source records attributed to Chi-Kwong Li.

104 records · Page 6Linked to original sources

arXiv2011

A note on the realignment criterion

For a quantum state in a bipartite system represented as a density matrix, researchers used the realignment matrix and functions on its singular values to study the separability of the quantum state. We obtain bounds for elementary symmetric functions of singular values of realignment matrices. This answers some open problems proposed by Lupo, Aniello, and Scardicchio. As a consequence, we show that the proposed scheme by these authors for testing separability would not work if the two subsystems of the bipartite system have the same dimension.

quant-ph↗

arXiv2011

Recursive Encoding and Decoding of Noiseless Subsystem and Decoherence Free Subspace

When the environmental disturbace to a quantum system has a wavelength much larger than the system size, all qubits localized within a small area are under action of the same error operators. Noiseless subsystem and decoherence free subspace are known to correct such collective errors. We construct simple quantum circuits, which implement these collective error correction codes, for a small number $n$ of physical qubits. A single logical qubit is encoded with $n=3$ and $n=4$, while two logical qubits are encoded with $n=5$. The recursive relations among the subspaces employed in noiseless subsystem and decoherence free subspace play essential rôles in our implementation. The recursive relations also show that the number of gates required to encode $m$ logical qubits increases linearly in $m$.

quant-ph↗

arXiv2011

The automorphism group of separable states in quantum information theory

We show that the linear group of automorphism of Hermitian matrices which preserves the set of separable states is generated by \emph{natural} automorphisms: change of an orthonormal basis in each tensor factor, partial transpose in each tensor factor, and interchanging two tensor factors of the same dimension. We apply our results to preservers of the product numerical range.

quant-ph↗

arXiv2010

Interpolation problems by completely positive maps

Given commuting families of Hermitian matrices {A1, ..., Ak} and {B1, ...., Bk}, conditions for the existence of a completely positive map L, such that L(Aj) = Bj for j = 1, ...,k, are studied. Additional properties such as unital or / and trace preserving on the map ? are also considered. Connections of the study to dilation theory, matrix inequalities, unitary orbits, and quantum information science are mentioned.

math.FA↗

arXiv2010

Higher rank numerical ranges of normal matrices

The higher rank numerical range is closely connected to the construction of quantum error correction code for a noisy quantum channel. It is known that if a normal matrix $A \in M_n$ has eigenvalues $a_1, \..., a_n$, then its higher rank numerical range $Λ_k(A)$ is the intersection of convex polygons with vertices $a_{j_1}, \..., a_{j_{n-k+1}}$, where $1 \le j_1 < \... < j_{n-k+1} \le n$. In this paper, it is shown that the higher rank numerical range of a normal matrix with $m$ distinct eigenvalues can be written as the intersection of no more than $\max\{m,4\}$ closed half planes. In addition, given a convex polygon ${\mathcal P}$ a construction is given for a normal matrix $A \in M_n$ with minimum $n$ such that $Λ_k(A) = {\mathcal P}$. In particular, if ${\mathcal P}$ has $p$ vertices, with $p \ge 3$, there is a normal matrix $A \in M_n$ with $n \le \max\left\{p+k-1, 2k+2 \right\}$ such that $Λ_k(A) = {\mathcal P}$.

math.FA↗

arXiv2010

Evolution of unconditional dispersal in periodic environments

Organisms modulate their fitness in heterogeneous environments by dispersing. Prior work shows that there is selection against "unconditional" dispersal in spatially heterogeneous environments. "Unconditional" means individuals disperse at a rate independent of their location. We prove that if within-patch fitness varies spatially and between two values temporally, then there is selection for unconditional dispersal: any evolutionarily stable strategy (ESS) or evolutionarily stable coalition (ESC) includes a dispersive phenotype. Moreover, at this ESS or ESC, there is at least one sink patch (i.e. geometric mean of fitness less than one) and no sources patches (i.e. geometric mean of fitness greater than one). These results coupled with simulations suggest that spatial-temporal heterogeneity due to abiotic forcing result in either an ESS with a dispersive phenotype or an ESC with sedentary and dispersive phenotypes. In contrast, spatial-temporal heterogeneity due to biotic interactions can select for higher dispersal rates that ultimately spatially synchronize population dynamics.

q-bio.PE↗

arXiv2010

"Maps preserving the spectrum of generalized Jordan product of operators", and its "Addendum"

In the paper "Maps preserving the spectrum of generalized Jordan product of operators", we define a generalized Jordan products on standard operator algebras $A_1, A_2$ on complex Banach spaces $X_1, X_2$, respectively. This includes the usual Jordan product $A_1 \circ A_2 = A_1 A_2 + A_2 A_1$, and the triple $\{A_1,A_2,A_3\} = A_1 A_2 A_3 + A_3 A_2 A_1$. Let a map $Φ: A_1 \to A_2$ prserving the spectra of the products $$ σ(Φ(A_1) \circ ... \circ Φ(A_k)) = σ(A_1\circ ... \circ A_k) $$ whenever any one of $A_1, ..., A_k$ has rank at most one. It is shown in this paper that if the range of $Φ$ contains all operators of rank at most three, then $Φ$ must be a Jordan isomorphism multiplied by an $m$th root of unity. Similar results for maps between self-adjoint operators acting on Hilbert spaces are also obtained. After our paper "Maps preserving the spectrum of generalized Jordan product of operators" was published in Linear Algebra Appl. 432 (2010), 1049-1069, Jianlian Cui pointed out that some arguments in the proof of Theorem 3.1 are not entirely clear and accurate. Here we supply some details in the "Addendum".

math.FA↗

arXiv2008

Quantum error correction and generalized numerical ranges

For a noisy quantum channel, a quantum error correcting code exists if and only if the joint higher rank numerical ranges associated with the error operators of the channel is non-empty. In this paper, geometric properties of the joint higher rank numerical ranges are obtained and their implications to quantum computing are discussed. It is shown that if the dimension of the underlying Hilbert space of the quantum states is sufficiently large, the joint higher rank numerical range of operators is always star-shaped and contains a non-empty convex subset. In case the operators are infinite dimensional, the joint infinite rank numerical range of the operators is a convex set closely related to the joint essential numerical ranges of the operators.

math.FA↗

arXiv2008

Higher rank numerical ranges and low rank perturbations of quantum channels

For a positive integer $k$, the rank-$k$ numerical range $Λ_k(A)$ of an operator $A$ acting on a Hilbert space $\cH$ of dimension at least $k$ is the set of scalars $λ$ such that $PAP = λP$ for some rank $k$ orthogonal projection $P$. In this paper, a close connection between low rank perturbation of an operator $A$ and $Λ_k(A)$ is established. In particular, for $1 \le r < k$ it is shown that $Λ_k(A) \subseteq Λ_{k-r}(A+F)$ for any operator $F$ with $\rank (F) \le r$. In quantum computing, this result implies that a quantum channel with a $k$-dimensional error correcting code under a perturbation of rank $\le r$ will still have a $(k-r)$-dimensional error correcting code. Moreover, it is shown that if $A$ is normal or if the dimension of $A$ is finite, then $Λ_k(A)$ can be obtained as the intersection of $Λ_{k-r}(A+F)$ for a collection of rank $r$ operators $F$. Examples are given to show that the result fails if $A$ is a general operator. The closure and the interior of the convex set $Λ_k(A)$ are completely determined. Analogous results are obtained for $Λ_\infty(A)$ defined as the set of scalars $λ$ such that $PAP = λP$ for an infinite rank orthogonal projection $P$. It is shown that $Λ_\infty(A)$ is the intersection of all $Λ_k(A)$ for $k = 1, 2, >...$. If $A - μI$ is not compact for any $μ\in \IC$, then the closure and the interior of $Λ_\infty(A)$ coincide with those of the essential numerical range of $A$. The situation for the special case when $A-μI$ is compact for some $μ\in \IC$ is also studied.

math.FA↗

arXiv2008

Canonical forms, higher rank numerical range, convexity, totally isotropic subspace, matrix equations

Results on matrix canonical forms are used to give a complete description of the higher rank numerical range of matrices arising from the study of quantum error correction. It is shown that the set can be obtained as the intersection of closed half planes (of complex numbers). As a result, it is always a convex set in $\mathcal C$. Moreover, the higher rank numerical range of a normal matrix is a convex polygon determined by the eigenvalues. These two consequences confirm the conjectures of Choi et al. on the subject. In addition, the results are used to derive a formula for the optimal upper bound for the dimension of a totally isotropic subspace of a square matrix, and verify the solvability of certain matrix equations.

math.FA↗

arXiv2007

Condition for the higher rank numerical range to be non-empty

It is shown that the rank-$k$ numerical range of every $n$-by-$n$ complex matrix is non-empty if $n \ge 3k - 2$. The proof is based on a recent characterization of the rank-$k$ numerical range by Li and Sze, the Helly's theorem on compact convex sets, and some eigenvalue inequalities. In particular, the result implies that $Λ_2(A)$ is non-empty if $n \ge 4$. This confirms a conjecture of Choi et al. If $3k-2>n>0$, an $n$-by-$n$ complex matrix is given for which the rank-$k$ numerical range is empty. Extension of the result to bounded linear operators acting on an infinite dimensional Hilbert space is also discussed.

math.FA↗

arXiv2003

Eigenvalues, singular values, and Littlewood-Richardson coefficients

We characterize the relationship between the singular values of a complex Hermitian (resp., real symmetric, complex symmetric) matrix and the singular values of its off-diagonal block. We also characterize the eigenvalues of an Hermitian (or real symmetric) matrix C=A+B in terms of the combined list of eigenvalues of A and B. The answers are given by Horn-type linear inequalities. The proofs depend on a new inequality among Littlewood-Richardson coefficients.

math.AG↗

arXiv2001

Applications of Perron-Frobenius Theory to Population Dynamics

By the use of Perron-Frobenius theory, simple proofs are given of the Fundamental Theorem of Demography and of a theorem of Cushing and Yicang on the net reproductive rate occurring in matrix models of population dynamics. The latter result is further refined with some additional nonnegative matrix theory. When the fertility matrix is scaled by the net reproductive rate, the growth rate of the model is 1. More generally, we show how to achieve a given growth rate for the model by scaling the fertility matrix. Demographic interpretations of the results are given.

math.RA↗

arXiv1996

Isometries of cross products of sequence spaces

Let $X_0, X_1, ..., X_k$ with $k \in \IN\cup\{\infty\}$ be sequence spaces $($finite or infinite dimensional$)$ over $\IC$ or $\IR$ with absolute norms $N_i$ for $i = 0, ..., k$, $($i.e., with 1-unconditional bases$)$ such that $\dim X_0 = k$. Define an absolute norm on the cross product space $($also known as the $X_0$ 1-unconditional sum$)$ $X_1 \times ... \times X_k$ by $$ N(x_1, ..., x_k) = N_0(N_1(x_1), ..., N_k(x_k)) \quad \hbox{for all} \quad (x_1, ..., x_k) \in X_1 \times ... \times X_k. $$ We show that every sequence space with an absolute norm has an intrinsic cross product structure of this form. The result is used to prove a characterization of isometries of complex cross product spaces that covers all the existing results. We demonstrate by examples and the theory of finite reflection groups that it is impossible to extend the complex result to the real case. Nevertheless, some new isometry theorems are obtained for real cross product spaces.

math.FA↗