bibkey: erewgoldstein2025magic
authors: Muhammad Erew and Moshe Goldstein
year: 2025
title: 'Extremizing Measures of Magic on Pure States by Clifford-stabilizer States'
doi: 10.48550/arXiv.2512.19657
url: https://arxiv.org/abs/2512.19657v2
claim: The ququint state and Wigner convention are taken from the paper; the constrained direction analysis is left open there. Conjecture 1 asserts that every SIC-POVM fiducial state is a Clifford-stabilizer state (uniquely stabilized by a subgroup of the finite eigenphase-extended Clifford group).
strata_touched:
  - D5/S3/Quantum/Magic/QuquintWignerCriticalGeometry
  - D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation
license: citation-only
triage: anchor

Ququint Critical Geometry

Verified locator

DOI 10.48550/arXiv.2512.19657, https://arxiv.org/abs/2512.19657v2 and https://arxiv.org/html/2512.19657v2. Appendix E, equation (E.3a), gives the state (1,1,zeta^3,1,zeta^2)/sqrt(5). Equation (2.16) gives the phase-point convention after renaming the paper’s (p,q) to (q,p). Section 4.4.2, after (4.55), leaves the mana behavior for the constrained variations to the reader; the last row of Table 2 is undetermined.

The saved full text and abstract were read in the preceding implementation attempt and inspected again for this continuation. PR #5657 records the independent source check. The current Lean development certifies the exact geometry and vanishing first variation. QuquintStrictDecrease proves the exact normalized change and strict mana decrease along every nonzero direction in the specified constrained tangent family. This is a local repository result, not a claim that the source paper contains that proof.

This lane does not claim a general solution of mana extremisation, other dimensions, other critical points, that Claim C is the authors’ verbatim conjecture, or global novelty beyond the recorded search.

Conjecture 1 and its definitions

From the v2 TeX source main.tex, for a single qudit of prime dimension $d$ with $X|j⟩=|j+1⟩$ and $Z|j⟩={\omega }^j|j⟩$:

• Pauli group: “${\text{mP}}_{1,d}=\{(-1{)}^x{\zeta }^k{X}^a{Z}^b\mid a,b,k\in {\mathbb{Z}}_d,x\in {\mathbb{Z}}_2\}$, where $\zeta =e^{\pi i/d}$”.
• Clifford operations: “unitary transformations $C\in \mathrm{U}(\text{mH})$ that map the generalized Pauli group onto itself under conjugation”.
• Eigenphase-extended group: “${\text{mC}}_{N,d}^{\prime }=\left\{\lambda C\right|C\in {\stackrel{N,d,\lambda \in \Lambda (\backslash mC}{\text{mC}}}_{N,d})\}$, where $\Lambda ({\stackrel{N,d)\; denotes\; the\; set\; of\; all\; eigenvalues\; of\; all\; elements\; in\; \backslash mC}{\text{mC}}}_{N,d}$”, with ${\widetilde{\text{mC}}}_{N,d}$ the Clifford unitaries of determinant one.
• Clifford-stabilizer state: “A \emph{Clifford-stabilizer state} is a normalized pure state whose density operator is the projector onto a Clifford-stabilized subspace. Equivalently, it is the unique normalized state (up to an overall global phase) that spans a one-dimensional Clifford-stabilized space.”
• SIC fiducial: “a normalized state $|\psi ⟩\in {\text{mH}}_{1,d}$ such that $\left|⟨\psi |T_{\mathbf{\chi }}|\psi ⟩\right|=\frac{1}{\sqrt{d+1}}$ $\forall \mathbf{\chi }\in {\mathbb{V}}_{1,d}\setminus \{0\}$”.
• Conjecture 1 (Stabilizer nature of SIC fiducials): “Every SIC-POVM fiducial state is a Clifford-stabilizer state.”