A Phase-Space Geometric Measure of Magic in Qubit Systems

Abstract

Characterizing quantum magic -- the resource enabling computational advantage beyond stabilizer circuits -- is subtle in qubit systems because established measures can give conflicting information about the same state. We introduce C(rho), the l1 distance from a state's discrete Wigner function to the convex hull of stabilizer Wigner functions, and study its relationship to the stabilizer extent Gamma(rho) via the tightness ratio kappa(rho) := (Gamma(rho)-1)/C(rho). For three two-qubit families in the repetition-code subspace span{|00>,|11>}, we prove kappa takes exact integer values constant over each family: kappa=1 for the R_y and Bell+R_z families, kappa=2 for the R_x family. The factor-of-2 gap arises because imaginary coherence concentrates Wigner negativity at 2 of 16 phase-space points rather than 4, leaving Gamma unchanged. The optimal dual witnesses are logical Pauli operators of the repetition code, revealing that C is a fault-tolerant observable invariant under correctable errors -- an unexpected connection between phase-space geometry and quantum error correction. We prove a sharp bound Gamma >= 1 + C/M_n, establish a hemispheric dichotomy in tensor-product behavior where superadditivity of C fails for northern-hemisphere states with deficit approximately 0.335 C(rho), and show C is not a magic monotone under the full Clifford group, so asymptotic distillation rates require Gamma.