  Semidefinite programming optimises a linear objective function over a
spectrahedron, and is one of the major advances of mathematical optimisation.
Spectrahedra are described by linear pencils, which are linear matrix
polynomials with hermitian matrix coefficients. Our focus will be on
dimension-free linear pencils where the variables are permitted to be hermitian
matrices. A major question on linear pencils, and matrix theory in general, is
Kippenhahn's Conjecture. The conjecture states that given a linear pencil $xH +
yK $ if the hermitian matrices $H$ and $K$ generate the full matrix algebra,
then the pencil must have at least one simple eigenvalue for some $x$ and $y$.
The conjecture is known to be false, via a single counterexample due to Laffey.
A dimension-free version of the conjecture, known as the Quantum Kippenhahn
Theorem, has recently been proven true non-constructively. We present a novel
family of counterexamples to Kippenhahn's Conjecture, and use this family to
construct concrete examples of the Quantum Kippenhahn Theorem.
