The Lovász number is not alone: a continuum of Rényi points in the entanglement-assisted spectrum of graphs

Source abstract

Li and Zuiddam's entanglement-assisted asymptotic spectrum of graphs consists of the normalized graph parameters that are additive under disjoint unions, multiplicative under strong products and monotone under entanglement-assisted cohomomorphisms; the entanglement-assisted Shannon capacity is their pointwise minimum. To our knowledge, the Lovász number $\vartheta$ was its only known element. We find a continuum of others. Three quantities of Duan and Winter, $\log_2\vartheta$, the minimal Holevo capacity $C_{\min}$ and the logarithm of the no-signalling simulation cost $Σ$, are the smallest information radii of classical-quantum channels compatible with a graph, measured with Petz-Rényi divergences of orders $0$, $1$ and $2$. At order zero this is due to Schrijver (as credited by Dalai) and to Duan and Winter; at order one it is the min-max formula for the Holevo capacity. Interpolating in the order gives a family $\vartheta_α$, $0\leα\le2$, and we prove that every member lies in the spectrum. The main step, multiplicativity under strong products for channels that need not factor, follows from an exact nesting identity for compressions. Sandwiched Rényi radii of order $β\in[1/2,\infty]$ give the same family, at $α=2-1/β$. On the five-cycle the family takes every value in $[\sqrt5,5/2]$. A Hilbert-Schmidt inequality for the pentagon gives $C_{\min}(C_5)=\log_2(5/2)$, proving a conjecture of Duan and Winter. We also settle the two open links of their inequality chain: the graph-level no-signalling simulation rate always equals its single-letter value $\log_2Σ(G)$, and a 32-vertex graph from the Mermin-Peres magic square has $C_{\min}<\log_2Σ$, certified in exact rational arithmetic. Finally, $Σ$ never exceeds the projective rank of the complement; a rank inequality for the same square makes this strict on a 1440-vertex graph.