Construct a measure of magic from Rényi multi-entropies

Construct a measure of quantum magic, meaning non-stabilizerness, solely from Rényi multi-entropies.

Background

The finite-field Gauss-sum framework in the paper relies on the quadratic phase structure of stabilizer states and yields Rényi multi-entropies that are rational multiples of log 2. The authors suggest that deviations from this behavior could witness non-stabilizerness, as illustrated by a three-qubit hypergraph state with a cubic phase.

Beyond using such deviations as a possible witness, the paper raises the more ambitious problem of designing a quantitative measure of magic from the Rényi multi-entropies. This problem is explicitly left unresolved.