Generalized interlacing inequalities

Li, Chi Kwong; Poon, Yiu‐Tung; Sze, Nung-Sing

doi:10.1080/03081087.2011.619534

Cited by 3 publications

(2 citation statements)

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“…The study of Λ p,q (A) was motivated by the search of quantum error correction code for a given quantum channel, for example, see [13,17]. In [13] it was shown that if the dimension n is sufficiently large, then the complex joint (p, q)matricial range is non-empty and star-shaped.…”

Section: Introductionmentioning

confidence: 99%

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Joint matricial range and joint congruence matricial range of operators

Lau

Poon

et al. 2020

Adv. Oper. Theory

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Let A = (A 1 , . . . , A m ), where A 1 , . . . , A m are n × n real matrices. The real joint (p, q)-matricial range of A, Λ R p,q (A), is the set of m-tuple of q × q real matrices (B 1 , . . . , B m ) such that (X * A 1 X, . . . , X * A m X) = (I p ⊗ B 1 , . . . , I p ⊗ B m ) for some real n × pq matrix X satisfying X * X = I pq . It is shown that if n is sufficiently large, then the set Λ R p,q (A) is non-empty and star-shaped. The result is extended to bounded linear operators acting on a real Hilbert space H, and used to show that the joint essential (p, q)-matricial range of A is always compact, convex, and non-empty. Similar results for the joint congruence matricial ranges on complex operators are also obtained.

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Section: Introductionmentioning

confidence: 99%

“…In [13] it was shown that if the dimension n is sufficiently large, then the complex joint (p, q)matricial range is non-empty and star-shaped. Consequently, if the underlying Hilbert space of a quantum system has sufficiently high dimension, one can always find a correctable subsystem for a quantum channel; see [6] and [17] more details.…”

Section: Introductionmentioning

confidence: 99%

Joint matricial range and joint congruence matricial range of operators

Lau

Poon

et al. 2020

Adv. Oper. Theory

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Convexity and star-shapedness of matricial range

Lau

Poon

et al. 2018

Journal of Functional Analysis

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Let A = (A 1 , . . . , A m ) be an m-tuple of bounded linear operators acting on a Hilbert space H. Their joint (p, q)-matricial range Λ p,q (A) is the collection of (B 1 , . . . , B m ) ∈ M m q , where I p ⊗ 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 thaton a closed subspace of H, and consider the joint essential (p, q)-matricial range Λ ess p,q (A) = {cl(Λ p,q (A 1 + F 1 , . . . , A m + F m )) : F 1 , . . . , F m are compact operators}.Both sets are shown to be convex, and the latter one is always non-empty and compact.Keywords: Joint matricial range, Joint essential numerical range, Higher rank numerical range, Bounded linear operators 2010 MSC: 47A12, 47A13, 47A55, 15A60

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Absolutely k -incoherent quantum states and spectral inequalities for the factor width of a matrix

et al. 2022

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A matrix is said to have factor width at most k if it can be written as a sum of positive semidefinite matrices that are non-zero only in a single k ˆk principal submatrix. We explore the "factor-width-k rank" of a matrix, which is the minimum number of rank-1 matrices that can be used in such a factorwidth-at-most-k decomposition. We show that the factor width rank of a banded or arrowhead matrix equals its usual rank, but for other matrices they can differ. We also establish several bounds on the factor width rank of a matrix, including a tight connection between factor-width-k rank and the k-clique covering number of a graph, and we discuss how the factor width and factor width rank change when taking Hadamard products and Hadamard powers.

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Generalized interlacing inequalities

Abstract: We discuss some applications of generalized interlacing inequalities of Ky Fan to the study of (a) some classical matrix inequalities and (b) matrix problems in quantum information science.

Cited by 3 publications

References 20 publications

Joint matricial range and joint congruence matricial range of operators

Joint matricial range and joint congruence matricial range of operators

Convexity and star-shapedness of matricial range

Absolutely k -incoherent quantum states and spectral inequalities for the factor width of a matrix

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