Higher rank numerical ranges of normal matrices

Hwa Long Gau, Chi Kwong Li, Yiu Tung Poon, Nung Sing Sze

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

Abstract

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 ε Mn has eigenvalues a1, ⋯ , a n, then its higher rank numerical range Γκ(A) is the intersection of convex polygons with vertices aj1 , ⋯ , ajn-k+1, where 1 ≤ j1 < ⋯ ≤ j n-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 ε Mn with minimum n such that Δκ(A) = P. In particular, if P has p vertices, with p ≥ 3, there is a normal matrix A ε Mn with n ≤ max {p + k -1, 2k + 2} such that Γκ(A) = P.

Original languageEnglish
Pages (from-to)23-43
Number of pages21
JournalSIAM Journal on Matrix Analysis and Applications
Volume32
Issue number1
DOIs
StatePublished - 2011

Keywords

  • Convex polygon
  • Higher rank numerical range
  • Normal matrices
  • Quantum error correction

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