Relationship between the optimized witness and stabilizer Rényi entropy

Determine whether the optimized Wigner-function witness $M$ and the stabilizer Rényi entropy $\mathcal{M}_2$ capture complementary resources or partially overlapping resources as measures of non-stabilizerness.

Background

The paper compares the optimized Wigner-function witness M, which measures anisotropic concentration of probabilities across a complete set of stabilizer MUBs, with the stabilizer Rényi entropy M2\mathcal{M}_2, which is Clifford invariant and measures global phase-space correlations.

Although the two quantities show qualitatively similar behavior along interpolations between stabilizer and Haar-random states, they select different extremal states: SIC states maximize the stabilizer Rényi entropy in the cited studies, whereas the witness M is maximized by states with a dominant component in each MUB. The authors explicitly leave unresolved whether the two quantities represent complementary resources or partially overlapping aspects of non-stabilizerness.

References

Thus, whether (\ref{R}) and (\ref{M}) capture complementary or partially overlapping resources remains an open question.

— Optimized discrete Wigner representations and non-stabilizerness in qubit systems  (2610.10471 - Klimov et al., 7 Oct 2026) in Section IV, subsection “Comparison with the stabilizer Rényi entropy”