Manuscript Oai365-A@1 · Oai365-A.tex · proposition · a3840e3d

Proposition: Finite-generation obstruction

Quantum-process data and the event carrier › Faithful generated affine representations · lines 334–354 · proof included or linked

Oai365-A@1#prop:finite

A finite event carrier cannot satisfy (eq:generated) with $K$ linearly isomorphic to

L_{m} = {(t, x) \in R \times R^{m - 1} : t \geq ∥ x ∥_{2}}, m \geq 3.

Proof. A finitely generated cone is closed and polyhedral and has only finitely many extreme rays. The Lorentz cone has the distinct extreme rays

R_{\geq 0} (1, u), u \in S^{m - 2} .

For example, the nonnegative functional $(t, x) \mapsto t - u \cdot x$ vanishes on exactly that ray. Invertible linear maps preserve extreme rays.

Displayed formulas in this passage

unlabeled · line 338
$L_{m} = {(t, x) \in R \times R^{m - 1} : t \geq ∥ x ∥_{2}}, m \geq 3.$
unlabeled · line 348
$R_{\geq 0} (1, u), u \in S^{m - 2} .$