The joint numerical range of three hermitian $4\times 4$ matrices

TL;DR

This paper studies the geometric structure of the joint numerical range of three hermitian 4x4 matrices, classifying boundary features and exploring implications for quantum entanglement.

Contribution

It classifies possible boundary structures of the joint numerical range and introduces the separable numerical range in the context of quantum entanglement.

Findings

Identified 15 classes of boundary structures for the joint numerical range.

Proved that the intersection of three distinct one-dimensional faces is a corner point.

Compared the boundary of the separable numerical range with that of the joint numerical range.

Abstract

We analyze the joint numerical range $W$ of three hermitian matrices of order four. In the generic case, this three-dimensional convex set has a smooth boundary. We analyze non-generic structures. Fifteen possible classes regarding the numbers of non-elliptic faces in the boundary of $W$ are identified and an explicit example is presented for each class. Secondly, it is shown that a nonempty intersection of three mutually distinct one-dimensional faces is a corner point. Thirdly, introducing a tensor product structure into $\mathbb C^4=\mathbb C^2\otimes\mathbb C^2$, one defines the separable joint numerical range - a subset of $W$ useful in studies of quantum entanglement. The boundary of the separable numerical range is compared with that of $W$.