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

28 pages, 5 figures; version v2 extends version v1 in applications and examples and has more detailed proofs. Any comments are welcomed

References in corpus (3)

• Observing geometry of quantum states in a three-level system
• Symmetry Breaking and Convex Set Phase Diagrams for the q-state Potts Model
• A new signature of quantum phase transitions from the numerical range

Cited by in corpus (1)

• Matrix systems, algebras, and open maps