Optimized discrete Wigner representations and non-stabilizerness in qubit systems

Abstract

The apparent absence of a stabilizer-speci?c phase-space description for qubit systems is not fundamental, but arises from ?xing a state-independent discrete Wigner map. Allowing the phase-space representation to adapt to the state through a simple optimization procedure, we prove that every qubit stabilizer state admits an exact delta-function discrete Wigner representation for any ?xed phase-space partition. This optimization can be implemented directly at the level of measurement statistics by selecting maximal probabilities across mutually unbiased bases forming the partition. This construction naturally leads to a non-stabilizer witness de?ned by the optimized value of the Wigner function at the origin. Stabilizer states saturate the universal lower bound of this witness for pure states, whereas a value strictly larger than the bound certi?es non-stabilizerness. Although the witness is not a ?nite-size resource monotone, it becomes sharply concentrated and e?ectively invariant under random Cli?ord transformations in the large-qubit limit, suggesting an asymptotically linear growth law for broad families of non-stabilizer states. Our results restore a meaningful geometric phase-space characterization of qubit stabilizer structure and provide an operational diagnostic of non-stabilizerness.