Kubo-Ando Means and Rigidity of Quantum Positivity Cones

We investigate the stability of quantum positivity cones under nonlinear operator means. Specifically, we examine how Kubo–Ando means interact with the separable, positive partial transpose (PPT), and Schmidt-number cones. By analyzing the curvature of operator monotone functions at the identity, we give a strict rigidity phenomenon: weighted arithmetic means are the only Kubo–Ando means that preserve the separable cone in all dimensions. We show that the strictly positive curvature of any non-arithmetic mean explicitly forces a violation of the PPT condition, even in the foundational two-qubit setting, and can strictly increase the Schmidt number of the resulting operator. Finally, using the Choi–Jamiołkowski correspondence, we translate these geometric obstructions to the map-theoretic setting, concluding that convex mixing is the uniquely permissible Kubo–Ando operation for preserving entanglement-breaking quantum channels.