Non-Subhomogeneity of Minimal Operator Systems over Positive Semidefinite and Lorentz Cones

Abstract

The minimal operator systems over ${\mathrm{Mat}}_k(\mathbb{C}{)}_{+}$, $k⩾2$, and the Lorentz cones ${\mathcal{L}}_m$, $m⩾4$, are not subhomogeneous. For the $2\times 2$ cone we construct extreme positive maps of arbitrarily large output dimension. Positive retracts give the remaining cases. Equivalently, for fixed $k⩾2$ and $d$, there exist $s>d$ and an entangled positive operator on ${\mathbb{C}}^k\otimes {\mathbb{C}}^s$ such that every compression of the second factor to ${\mathbb{C}}^d$ is separable.

The results were produced almost entirely by artificial intelligence. I posed questions and suggested approaches, clarified and simplified the arguments, checked them for correctness, and edited the final text. There is no settled convention for attributing such work. I do not claim ownership and will not submit it to a journal. Comments and corrections are very welcome