Develop an entanglement-assisted theory of probabilistic refinements

Establish whether the entanglement-assisted graph preorder admits a probabilistic-refinement theory analogous to the log-convex interpolation theory for the ordinary asymptotic spectrum, and determine whether the Rényi points \(\vartheta_\alpha\) arise from such interpolations.

Background

The ordinary asymptotic spectrum has a probabilistic-refinement construction that produces log-convex combinations of spectral points. The authors ask whether an analogous construction exists for the entanglement-assisted preorder.

They explain that, even if such a theory exists, the order-one point would not be an interpolation between the Lovász number and $\Sigma$ under the expected vertex-transitive evaluation rule; the status of the points with $\alpha<1/2$ is unknown.