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arXiv · 1907.04768

Kippenhahn's Theorem for joint numerical ranges and quantum states

Daniel Plaumann·Rainer Sinn·Stephan Weis
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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.

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Daniel Plaumann, Rainer Sinn, Stephan Weis. 2019-07-10. Kippenhahn's Theorem for joint numerical ranges and quantum states. https://doi.org/10.1137/19m1286578

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