Elliptical higher rank numerical range of some Toeplitz matrices

AbstractIn this article the rank-k numerical range ∧k (A) of an entrywise nonnegative matrix A is investigated. Extending the notion of elements of maximum modulus in ∧k (A), we examine their location on the complex plane. Further, an application of this theory to ∧k (L(λ)) of a Perron polynomial L(λ) is elaborated via its companion matrix C L.

Download Full-text

On some reciprocal matrices with elliptical components of their Kippenhahn curves

Special Matrices ◽

10.1515/spma-2021-0151 ◽

2021 ◽

Vol 10 (1) ◽

pp. 117-130

Author(s):

Muyan Jiang ◽

Ilya M. Spitkovsky

Keyword(s):

Numerical Range ◽

Main Diagonal ◽

Higher Rank

Abstract By definition, reciprocal matrices are tridiagonal n-by-n matrices A with constant main diagonal and such that ai , i +1 ai +1, i = 1 for i = 1, . . ., n − 1. We establish some properties of the numerical range generating curves C(A) (also called Kippenhahn curves) of such matrices, in particular concerning the location of their elliptical components. For n ≤ 6, in particular, we describe completely the cases when C(A) consist entirely of ellipses. As a corollary, we also provide a complete description of higher rank numerical ranges when these criteria are met.

Download Full-text

Multiaccess quantum communication and product higher rank numerical range

Quantum Information and Computation ◽

10.26421/qic13.7-8-1 ◽

2013 ◽

Vol 13 (7&8) ◽

pp. 541-566

Author(s):

Maciej Demianowicz ◽

Pawel Horodecki ◽

Karol Zyczkowski

Keyword(s):

Multiple Access ◽

Quantum Communication ◽

Numerical Range ◽

Analytical Techniques ◽

Noise Model ◽

Error Correction Codes ◽

Multiple Access Channels ◽

Exact Shape ◽

Higher Rank ◽

Quantum Error

In the present paper we initiate the study of the product higher rank numerical range. The latter, being a variant of the higher rank numerical range [M.--D. Choi {\it et al.}, Rep. Math. Phys. {\bf 58}, 77 (2006); Lin. Alg. Appl. {\bf 418}, 828 (2006)], is a natural tool for studying a construction of quantum error correction codes for multiple access channels. We review properties of this set and relate it to other numerical ranges, which were recently introduced in the literature. Further, the concept is applied to the construction of codes for bi--unitary two--access channels with a hermitian noise model. Analytical techniques for both outerbounding the product higher rank numerical range and determining its exact shape are developed for this case. Finally, the reverse problem of constructing a noise model for a given product range is considered.

Download Full-text