# Kippenhahn's Theorem for joint numerical ranges and quantum states

## TL;DR

This paper extends Kippenhahn's Theorem to joint numerical ranges of Hermitian matrices, providing a geometric framework for analyzing quantum states through convex hulls of semi-algebraic sets.

## Contribution

It introduces a new geometric approach to study joint numerical ranges and dual cones, linking algebraic curves with quantum state analysis.

## Key findings

- Joint numerical range is the convex hull of a semi-algebraic set.
- Bases of dual cones are closed under linear operations.
- Provides a new geometric method for quantum state analysis.

## Abstract

Kippenhahn's Theorem asserts that the numerical range of a matrix is the convex hull of a certain algebraic curve. Here, we show that the joint numerical range of finitely many Hermitian matrices is similarly the convex hull of a semi-algebraic set. We discuss an analogous statement regarding the dual convex cone to a hyperbolicity cone and prove that the class of bases of these dual cones is closed under linear operations. The result offers a new geometric method to analyze quantum states.