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

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 ${\mathbb{R}}^3$. Generically we find that $W(F)$ is 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.

arXiv comment23 pages, 4 figures. This version includes some minor improvements which could not be taken into account in the proofs of the published version

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• Bounding the joint numerical range of Pauli strings by graph parameters
• Kippenhahn's Theorem for joint numerical ranges and quantum states
• Geometry of entanglement and separability in Hilbert subspaces of dimension up to three
• On a theorem by Kippenhahn
• Uncertainty relations and joint numerical ranges
• An Effective Way to Determine the Separability of Quantum State
• Joint separable numerical range and bipartite entanglement witness
• Wigner distributions for n arbitrary observables
• Universal witnesses of vanishing energy gap