A Phase-Space Geometric Measure of Magic in Qubit Systems

TL;DR

This paper introduces a new phase-space geometric measure of quantum magic in qubit systems, analyzes its relationship with stabilizer extent, and reveals connections to quantum error correction and resource theory.

Contribution

It defines the measure C(rho), studies its relationship with Gamma(rho), and uncovers its invariance properties and limitations as a magic monotone.

Findings

kappa takes exact integer values for specific two-qubit families

C is a fault-tolerant observable invariant under correctable errors

C is not a magic monotone under the full Clifford group

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.