Classification of joint numerical ranges of three hermitian matrices of size three

Abstract

Abstract. The joint numerical range W (F ) of three hermitian 3-by-3 matrices F = (F 1 , F 2 , F 3 ) is a convex and compact subset in R 3 . We show that W (F ) is generically a three-dimensional oval. Assuming dim(W (F )) = 3, every one-or two-dimensional face of W (F ) is a segment or a filled ellipse. We prove that only ten configurations of these segments and ellipses are possible. We identify a triple F for each class and illustrate W (F ) using random matrices and dual varieties.

Hyperpolarization and the Physical Boundary of Liouville Space

“…", etc. Surprisingly, these natural questions are rarely posed in the NMR world, although they have not escaped the attention of mathematical physicists and applied mathematicians (Byrd and Khaneja (2003); Kimura and Kossakowski (2005); Bengtsson and Zyczkowski (2006); Goyal et al (2016); Szymański et al (2018)).…”

Quaternion matrix decomposition and its theoretical implications

“…Joint numerical range also has as wide applications as that of numerical range, and theoretically, it has a close connection to other fundamental results such as S-Lemma, which will be discussed in the next section. On the other hand, people generalized the results of joint numerical ranges in various ways [1,8,23,36]. In this section, we show how to further extend the classical results on the convexity of the joint numerical ranges [4,27] to the quaternion domain via the rank-one decomposition of quaternion matrices studied in the last section.…”

Hyperpolarization and the physical boundary of Liouville space

“…The map still needs to be completed by delineating the physical bounds on coherences and on operators for multiple-spin systems, including those that are not exchange-symmetric. There has already been significant progress in that direction (Goyal et al, 2016;Szymański et al, 2018).…”

“…Surprisingly, these natural questions are rarely posed in the NMR world, although they have not escaped the attention of mathematical physicists and applied mathematicians (Byrd and Khaneja, 2003;Kimura and Kos-Published by Copernicus Publications on behalf of the Groupement AMPERE. sakowski, 2005; Bengtsson and Zyczkowski, 2006;Goyal et al, 2016;Szymański et al, 2018).…”