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arXiv:2608.24648

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

Tim Netzer

math.OAquant-phSubmitted Aug 25, 2026• Announced Aug 26, 2026• 0 citations
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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.

Comments
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

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The minimal operator systems over Matk(C)+{\rm Mat}_k(\mathbb C)_+Matk​(C)+​, k⩾2k\geqslant2k⩾2, and the Lorentz cones Lm\mathcal L_mLm​, m⩾4m\geqslant4m⩾4, are not subhomogeneous. For the 2×22\times22×2 cone we construct extreme positive maps of arbitrarily large output dimension. Positive retracts give the remaining cases. Equivalently, for fixed k⩾2k\geqslant2k⩾2 and ddd, there exist s>ds>ds>d and an entangled positive operator on Ck⊗Cs\mathbb C^k\otimes\mathbb C^sCk⊗Cs such that every compression of the second factor to Cd\mathbb C^dCd is separable.
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There are various notions of positivity for matrices and linear matrix-valued maps that play important roles in quantum information theory. The cones of positive semidefinite matrices and completely positive linear maps, which represent quantum states and quantum channels respectively, are the most ubiquitous positive cones. There are also many natural cones that can been regarded as "more" or "less" positive than these standard examples. In particular, entanglement theory deals with the cones of separable operators and entanglement witnesses, which satisfy very strong and weak positivity properties respectively. Rather complementary to the various cones that arise in entanglement theory are norms. The trace and operator norms for operators and the diamond and completely bounded norms for superoperators are the typical norms that are seen throughout quantum information theory. In this work our main goal is to develop a family of norms that play a role analogous to the cone of entanglement witnesses. We investigate the basic mathematical properties of these norms, including their relationships with other well-known norms, their isometry groups, and their dual norms. We also make the place of these norms in entanglement theory rigorous by showing that entanglement witnesses arise from minimal operator systems, and analogously our norms arise from minimal operator spaces. Finally, we connect the various cones and norms considered here to several seemingly unrelated problems from other areas. We characterize the problem of whether or not non-positive partial transpose bound entangled states exist in terms of one of our norms, and provide evidence in favour of their existence. We also characterize the minimum gate fidelity of a quantum channel, the maximum output purity and its completely bounded counterpart, and the geometric measure of entanglement in terms of these norms.
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