2009/02/28 by Hwa-Long Gau, Chi-Kwong Li, Yiu-Tung Poon +1
Computer Science · Mathematics · Physics and Astronomy · #Coding theory and cryptography #Eigenvalues and eigenvectors #Intersection (aeronautics) #Matrix (chemical analysis) #Normal matrix #Numerical range #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Range (aeronautics) #Rank (graph theory) #Regular polygon #math-ph #math.FA #math.MP #msc:15A60 #msc:15A90 #msc:47N50 #msc:81P68 #quant-ph
paper · pdf · doi:10.1137/09076430x
published as SIAM J. Matrix Analysis Appl, 32:23-43, 2011 · 12 pages, 9 figures, to appear in SIAM Journal on Matrix Analysis and Applications
arxiv created 2010/11/02 · openalex publication_date 2011/01/01 · arxiv updated 2011/02/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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∈ Mn has eigenvalues a1,…,an, then its higher rank numerical range Λk(A) is the intersection of convex polygons with vertices aj1,…,a_jn-k+1, where 1≤ j1<…<jn-k+1≤ 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 P, a construction is given for a normal matrix A∈ Mn with minimum n such that Λk(A)=P. In particular, if P has p vertices, with p≥3, there is a normal matrix A∈ Mn with n≤max \p+k-1,2k+2\ such that Λk(A)=P.