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.

Figures (7)

• Figure 1: Numerical analysis of the hemispheric dichotomy for $\rho_T \otimes \sigma$ as $\sigma$ varies over the Bloch sphere. Top left: The deficit $C(\rho_T)+C(\sigma)-C(\rho_T\otimes\sigma)$ (red) and the improvement from entangled stabilizer vertices (blue) as a function of $\langle Z\rangle_\sigma$. Superadditivity fails precisely for $\langle Z\rangle_\sigma > 0$ (northern hemisphere, shaded red) and holds for $\langle Z\rangle_\sigma \leq 0$ (southern hemisphere and equator, shaded green). Top right: The deficit scales linearly with $C(\rho)$ (slope $\approx 0.335$, $R^2=0.977$), suggesting the factored form $\mathrm{deficit}(\rho,\sigma) = C(\rho)\cdot f(\langle Z\rangle_\sigma)$. Bottom: Universality across different first states $\rho$ (all magic states show the same dichotomy pattern), and the comparison between the full Wigner polytope and the product-only polytope.
• Figure 2: Preparation circuits for the three families in the [[2,1,1]] repetition-code subspace $\mathcal{C} = \mathrm{span}\{|00\rangle,|11\rangle\}$. The CNOT encodes the single-qubit state into the codespace. The rotation axis determines the Bloch-sphere orbit and hence $\kappa$: real coherence ($R_y$, XZ meridian, blue) gives $\kappa=1$; imaginary coherence ($R_x$, YZ meridian, red) gives $\kappa=2$; equatorial phase (Bell$+R_z$, equatorial circle, green) gives $\kappa=1$ with a piecewise-constant witness. Exact values of $C$ and $\Gamma$ are given in \ref{['thm:kappa_ry_rx', 'thm:kappa_BRz']}.
• Figure 3: The logical Bloch sphere for the repetition-code subspace $\mathcal{C} = \mathrm{span}\{|00\rangle,|11\rangle\}$. The inscribed octahedron has the six stabilizer states as vertices. The three family orbits are shown: $R_y$ (green, XZ meridian, $\kappa=1$), $R_x$ (red, YZ meridian, $\kappa=2$), and Bell$+R_z$ (orange, equatorial circle, $\kappa=1$). The $\kappa$ value is determined by the coherence direction, not the position along the orbit.
• Figure 4: Discrete Wigner functions of the $R_y$ (left) and $R_x$ (right) families at $\theta = \pi/3$. The $R_y$ state has four negative entries spread across phase space; the $R_x$ state has only two. The factor-of-2 difference in the number of negative entries directly explains $\kappa^{Ry} = 1$ vs $\kappa^{Rx} = 2$.
• Figure 5: $C(\theta)$, $\Gamma(\theta)-1$, and $\kappa(\theta)$ as functions of $\theta$ for the $R_y$ (left panel, $\kappa \equiv 1$) and $R_x$ (right panel, $\kappa \equiv 2$) families. The flatness of $\kappa$ across all $\theta$ is a non-trivial feature proved analytically in \ref{['thm:kappa_ry_rx']}.

Theorems & Definitions (41)

• Remark 2.1: Why the Wigner function, not P or Q
• Definition 3.1: Wigner distance
• Theorem 3.2: Structural properties of $C$
• Theorem 3.3: Dual representation, \ref{['app:duality']}
• Theorem 3.4: Simulation bound
• proof
• Definition 3.5: Tightness ratio
• Remark 3.6: $\kappa$ is undefined for stabilizer states
• Lemma 4.2: Wigner factorization
• proof