Exact inradius of the qubit stabilizer polytope

Determine the exact Hilbert–Schmidt inradius of the n-qubit stabilizer polytope, equivalently close the gap between the Triangle Criterion boundary purity threshold 1/(d-1/2) and the proven stabilizerness threshold 1/(d-a_*) with a_*=0.458327⋯, thereby completely determining the purity threshold for the onset of quantum magic.

Background

The paper establishes lower and upper bounds on the Hilbert–Schmidt inradius of the n-qubit stabilizer polytope, where d=2^n. Its result guarantees that every state with purity at most 1/(d-a_) is stabilizer-free of magic, with a_=0.458327⋯, while the Triangle Criterion provides a boundary construction at purity 1/(d-1/2). These bounds differ by a small dimension-independent constant gap.

Determining the exact inradius would identify the precise purity threshold at which magic can first appear. The authors indicate that resolving the gap may require using the discrete structure of stabilizer states beyond the spectral information captured by their second- and third-moment analysis.