Extend the Rényi family and product theorem to non-commutative graphs

Determine whether the optimized Rényi graph parameters \(\vartheta_\alpha\) extend to non-commutative graphs and whether their strong-product multiplicativity theorem remains valid in that setting.

Background

The paper treats finite classical graphs, whereas quantum Lovász-type parameters and entanglement-assisted preorders have non-commutative graph analogues.

For non-commutative graphs, the analogue of the equality between entanglement-assisted Shannon capacity and the quantum Lovász number is already known to fail, and fixed non-commutative bipartite graph simulation costs can be strictly submultiplicative. The authors ask whether the Rényi family and the graph-level product theorem nevertheless extend.