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

Abstract:The minimal operator systems over ${\rm Mat}_k(\mathbb C)_+$, $k\geqslant2$, and the Lorentz cones $\mathcal L_m$, $m\geqslant4$, are not subhomogeneous. For the $2\times2$ cone we construct extreme positive maps of arbitrarily large output dimension. Positive retracts give the remaining cases. Equivalently, for fixed $k\geqslant2$ 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

Operator Algebras (math.OA); Quantum Physics (quant-ph)

arXiv:2608.24648 [math.OA]

(or arXiv:2608.24648v1 [math.OA] for this version)

https://doi.org/10.48550/arXiv.2608.24648 arXiv-issued DOI via DataCite (pending registration)