Kippenhahn's Theorem for joint numerical ranges and quantum states
Algebraic Geometry
2021-09-28 v2 Operator Algebras
Quantum Physics
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.
Keywords
Cite
@article{arxiv.1907.04768,
title = {Kippenhahn's Theorem for joint numerical ranges and quantum states},
author = {Daniel Plaumann and Rainer Sinn and Stephan Weis},
journal= {arXiv preprint arXiv:1907.04768},
year = {2021}
}
Comments
28 pages, 5 figures; version v2 extends version v1 in applications and examples and has more detailed proofs. Any comments are welcomed