bibkey: dutta2026wignerdistance authors: Soumyojyoti Dutta; Tushar year: 2026 title: “A Phase-Space Geometric Measure of Magic in Qubit Systems” doi: 10.48550/arXiv.2603.20792 url: https://arxiv.org/abs/2603.20792v3 claim: “Conjecture 5.5 asks that for equatorial magic ρ and qubit σ with s(σ) > 0 the tensor deficit factor as C(ρ)·f(σ) for a universal f; Conjecture 5.6 asks for equatorial multiplicativity of the qubit Wigner distance; Conjecture 5.7 asks for self-tensor superadditivity on the nonpositive Bloch-product branch.” strata_touched:

• D5/S3/Quantum/Magic/QubitWignerDistanceTensorRules
• D5/S3/Quantum/Magic/QubitWignerDeficitFactorRefutation license: citation-only triage: anchor

A Phase-Space Geometric Measure of Magic in Qubit Systems

Soumyojyoti Dutta and Tushar, arXiv:2603.20792v3 [quant-ph]. Page numbers refer to the printed PDF. The frame is the Wootters product frame.

Page 3, Section 2.1:

The $n$-qubit Pauli group ${\mathcal{P}}_n$ consists of $n$-fold tensor products of $\{I,X,Y,Z\}$ with phases $\{\pm 1,\pm i\}$.

A stabilizer state is the unique $+1$ eigenstate of an abelian subgroup $S\le {\mathcal{P}}_n$ of size $2^n$.

Page 3, Section 2.3:

The single-qubit phase-point operators are indexed by ${\alpha }_k=(q_k,p_k)\in {\mathbb{F}}_2^2$: $A_{(q_k,p_k)}=\frac{1}{2}(I+(-1{)}^{p_k}X+(-1{)}^{q_k+p_k}Y+(-1{)}^{q_k}Z)$.

For $n$ qubits, $A_{\alpha }=A_{{\alpha }_1}\otimes \cdots \otimes A_{{\alpha }_n}$, and the discrete Wigner function is W_\rho(\alpha) = \frac{1}{2^n}\,\tr(\rho\,A_\alpha).

Page 3, Section 3.1 and Definition 3.1:

The stabilizer Wigner polytope is ${\mathcal{W}}_{\mathrm{free}}:=\mathrm{conv}\{W_{\sigma }:\sigma \in {\mathrm{Stab}}_n\}\subset {\mathbb{R}}^{4^n}$.

$C(\rho ):={\min }_{W_f\in {\mathcal{W}}_{\mathrm{free}}}\Vert W_{\rho }-W_f{\Vert }_1$.

Page 7, Observation 5.1:

For a single-qubit state $\rho$ with Bloch vector $\vec{r}$, write $s(\rho ):=\mathrm{sgn}(r_x{r}_y{r}_z)$.

Version 4 (arXiv:2603.20792v4, revised 6 October 2026), Section 5.2, Figure 1 caption and Section 5.4 (main.tex lines 1009, 1093 and 1146–1155); Conjecture 5.5 has the same wording in v3:

Let ${\rho }_T$ denote any equatorial single-qubit magic state;

Deficit $(1+C(\rho ))(1+C(\sigma ))-1-C(\rho \otimes \sigma )$ against $C(\rho )C(\sigma )$ for Haar-random pure single-qubit pairs

\begin{conjecture}[Factored deficit] For equatorial magic states $\rho$ and qubit states $\sigma$ with $s(\sigma )>0$: $\mathrm{deficit}(\rho ,\sigma )=C(\rho )\cdot f(\sigma )$ for a universal function $f\ge 0$ of the Pauli invariants $(|r_x|,|r_y|,|r_z|)$ of $\sigma$, vanishing iff $s(\sigma )\le 0$. \end{conjecture}

Page 8, Conjecture 5.6 (Equatorial multiplicativity):

For $⟨Z{⟩}_{\rho }=⟨Z{⟩}_{\sigma }=0$: $C(\rho \otimes \sigma )=C(\rho )+C(\sigma )+C(\rho )C(\sigma )$.

Page 8, Conjecture 5.7 (Self-tensor superadditivity, $s\le 0$ branch):

For any qubit state $\rho$ with $s(\rho )\le 0$: $C(\rho \otimes \rho )\ge 2C(\rho )$.

The encoding uses positive semidefinite complex matrices of trace one, Fin 2 × Fin 2 for phase points, and actual commuting subgroups of the matrix unitary group for stabilizer states. Unique eigenstates mean normalized eigenrays; their density matrices are vecMulVec ψ (star ψ). The real part in the Wigner transform agrees with its real-valued trace on Hermitian states. The distance is represented by the infimum of the same finite-coordinate absolute-value errors; WignerDistanceMinimum.COne_min and CTwo_min prove that this is the source minimum for every matrix. The equatorial proof constructs an attaining stabilizer mixture; the self-tensor proof bounds that minimum. The condition $s(\rho )\le 0$ is encoded as $r_x{r}_y{r}_z\le 0$, including the zero branch.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2603.20792
• URL: https://arxiv.org/abs/2603.20792v3
• URL: https://arxiv.org/abs/2603.20792v4 (latest version; Conjecture 5.5 and Conjectures 5.6, 5.7 unchanged)