2007/06/30 by Chi-Kwong Li, Nung-Sing Sze · 3 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Optimization Algorithms Research #Matrix Theory and Algorithms #Tensor decomposition and applications #math.FA #msc:15A21 #msc:15A24 #msc:15A60 #msc:15A90 #msc:81P68 #quant-ph
paper · pdf · doi:10.1090/s0002-9939-08-09536-1
published as Proc. Amer. Math. Soc., 136:3013-3023. 2008 · 10 pages. To appear in Proceedings of the American Mathematical Society
arxiv created 2008/01/24 · openalex publication_date 2008/04/30 · arxiv updated 2011/02/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
The 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 to verify the solvability of certain matrix equations.