Kippenhahn's Theorem for Joint Numerical Ranges and Quantum States

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 convex bases of these dual cones is closed under linear operations. The result offers a new geometric method to analyze quantum states.