For practitioners, a rough description of their argument is on pg 7 (note: separable=unentangled, β =1/kb T) of https://arxiv.org/pdf/2403.16850
This state [rho] is the Gibbs state at infinite temperature, and is in the interior of the convex hull of product states. So, as β tends to zero, ρ will eventually enter the interior of this convex hull, making it separable. This happens at a finite β which depends on system size.
This state [rho] is the Gibbs state at infinite temperature, and is in the interior of the convex hull of product states. So, as β tends to zero, ρ will eventually enter the interior of this convex hull, making it separable. This happens at a finite β which depends on system size.