A Phase-Space Geometric Measure of Magic in Qubit Systems

Abstract

Magic -- the resource enabling quantum computational advantage beyond stabilizer circuits -- has a clean phase-space characterization in odd prime dimensions that qubits notoriously lack. We study C(rho), the l_1 distance from a state's discrete Wigner function to the stabilizer polytope, and determine its exact geometry. We prove that the single-qubit Wigner l_1 metric has a cuboctahedral unit ball, that max_rho C(rho) = (sqrt(3)-1)/2 for a single qubit, attained precisely at the eight face states, and that C(rho_1 x ... x rho_n) <= prod_i (1 + C(rho_i)) - 1 for single-qubit factors. Together these give the exact tensor powers ((1+sqrt(3))/2)^n - 1 and the exact maximum of C over fully separable n-qubit states, leaving only entangled states open. Because no discrete Wigner function for qubits is Clifford covariant, any such measure is frame dependent, and we determine exactly which of its features are not. For a single qubit we show the valid frames are exactly eight, and that C is identical on all of them, so the maxima above are properties of the state rather than of the representation. For two qubits the conclusion reverses: enumerating all 6144 translation-covariant frames, they split into four equal classes on which the ratio C(rho_Rx)/C(rho_Ry) takes the values 1/2, 1 and 2. Within the Wootters frame we compute that structure exactly: a tetrahedral dichotomy governing when the product bound is saturated, and integer values 1, 2, 1 of the tightness ratio kappa := (Gamma-1)/C against the robustness of magic Gamma for three families in the [[2,1,1]] codespace, whose optimal witnesses are logical Pauli operators. We prove the bound Gamma >= 1 + C/M_n, and show C is not a magic monotone, so asymptotic distillation rates require Gamma.