Browsing by Author "Li, Chi-Kwong"

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HIGHER-RANK NUMERICAL RANGES AND DILATIONS
(2010-12-01)
Gau, Hwa-Long
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Li, Chi-Kwong
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Wu, Pei Yuan
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應用數學系
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Department of Applied Mathematics
For any n-by-n complex matrix A and any k, 1 <= k <= n, let Lambda(k)(A) = {lambda is an element of C : X*AX = lambda I(k) for some n-by-k X satisfying X*X = I(k)) be its rank-k numerical range. It is shown that if A is an n-by-n contraction, then Lambda(k)(A) = boolean AND{Lambda(k)(U) : U is an (n + d(A))-by-(n + d(A)) unitary dilation of A}, where d(A) = rank (I(n) - A* A). This extends and refines previous results of Choi and Li on constrained unitary dilations, and a result of Mirman on S(n)-matrices.
MINKOWSKI PRODUCT OF CONVEX SETS AND PRODUCT NUMERICAL RANGE
(2016-12)
Li, Chi-Kwong
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Pelejo, Diane Christine
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Poon, Yiu-Tung
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Wang, Kuo-Zhong
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應用數學系
Let K-1, K-2 be two compact convex sets in C. Their Minkowski product is the set K1K2 = {ab : a is an element of K-1, b is an element of K-2}. We show that the set K1K2 is star-haped if K-1 is a line segment or a circular disk. Examples for K-1 and K-2 are given so that K-1 and K-2 are triangles (including interior) and K1K2 is not star-shaped. This gives a negative answer to a conjecture by Puchala et. al concerning the product numerical range in the study of quantum information science. Additional results and open problems are presented.
MINKOWSKI PRODUCT OF CONVEX SETS AND PRODUCT NUMERICAL RANGE
(2016-12)
Li, Chi-Kwong
;
Pelejo, Diane Christine
;
Poon, Yiu-Tung
;
Wang, Kuo-Zhong
;
應用數學系
Let K-1, K-2 be two compact convex sets in C. Their Minkowski product is the set K1K2 = {ab : a is an element of K-1, b is an element of K-2}. We show that the set K1K2 is star-haped if K-1 is a line segment or a circular disk. Examples for K-1 and K-2 are given so that K-1 and K-2 are triangles (including interior) and K1K2 is not star-shaped. This gives a negative answer to a conjecture by Puchala et. al concerning the product numerical range in the study of quantum information science. Additional results and open problems are presented.
NUMERICAL RANGES OF THE PRODUCT OF OPERATORS
(2017-03-01)
Du, Hongke
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Li, Chi-Kwong
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Wang, Kuo-Zhong
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Wang, Yueqing
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Zuo, Ning
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.
OPTIMAL BOUNDS ON FUNCTIONS OF QUANTUM STATES UNDER QUANTUM CHANNELS
(2016-07)
Li, Chi-Kwong
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Pelejo, Diane Christine
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Wang, Kuo-Zhong
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應用數學系
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Department of Applied Mathematics
Let rho(1), rho(2) be quantum states and (rho(1), rho(2)) bar right arrow D(rho(1), rho(2)) be a scalar function such as the trace distance, the fidelity, and the relative entropy, etc. We determine optimal bounds for D(rho(1), Phi(rho(2))) for Phi is an element of S for different class of functions D(., .), where S is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels.
Product of operators and numerical range
(2016-01-02)
Chien, Mao-Ting
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Gau, Hwa-Long
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Li, Chi-Kwong
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Tsai, Ming-Cheng
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Wang, Kuo-Zhong
We show that a bounded linear operator A is an element of B(H) is a multiple of a unitary operator if and only if AZ and Z A always have the same numerical radius or the same numerical range for all (rank one) Z is an element of B(H). More generally, for any bounded linear operators A, B is an element of B(H), we show that AZ and ZB always have the same numerical radius (resp., the same numerical range) for all (rank one) Z is an element of B(H) if and only if A = e(it) B (resp., A = B) is a multiple of a unitary operator for some t is an element of [0, 2 pi). We extend the result to other types of generalized numerical ranges including the k-numerical range and the higher rank numerical range.
Product of two positive contractions
(2016-07-15)
Li, Chi-Kwong
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Pelejo, Diane Christine
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Wang, Kuo-Zhong
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應用數學系
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Department of Applied Mathematics
Several characterizations are given for a square matrix that can be written as the product of two positive (semidefinite) contractions. Based on one of these characterizations, and the theory of alternating projections, a Matlab program is written to check the condition and construct the two positive contractions whose product equal to the given matrix, if they exist. (C) 2016 Elsevier Inc. All rights reserved.
The spectrum of the product of operators, and the product of their numerical ranges
(2015-03-15)
Li, Chi-Kwong
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Tsai, Ming-Cheng
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Wang, Kuo-Zhong
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Wong, Ngai-Ching
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應用數學系
We show that a compact operator A is a multiple of a positive semi-definite operator if and only if sigma(AB) subset of <(W(A)W(B))over bar>, for all (rank one) operators B. An example of a normal operator is given to show that the equivalence conditions may fail in general. We then obtain conditions to identify other classes of operators A so that equivalence conditions hold. (C) 2014 Elsevier Inc. All rights reserved.