slug: erew-goldstein-2025-sic-clifford-stabilizer bibkey: erewgoldstein2025magic doi: 10.48550/arXiv.2512.19657 url: https://arxiv.org/abs/2512.19657v2 triage: theorem motivation_gids:

• D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.result

A qutrit SIC fiducial that is not a Clifford-stabilizer state

Problem

M. Erew and M. Goldstein, Extremizing Measures of Magic on Pure States by Clifford-stabilizer States, arXiv:2512.19657v2, prove that states uniquely stabilized by a subgroup of the finite eigenphase-extended Clifford group extremize several magic measures, and state:

Conjecture 1 (Stabilizer nature of SIC fiducials). Every SIC-POVM fiducial state is a Clifford-stabilizer state.

The verbatim definitions are in the literature note. Issue #14173 reads it for every prime $d$ and every normalized single-qudit fiducial.

Motivation

The conjecture would explain the extremality of SIC fiducials for $L^p$-norm magic and stabilizer Rényi entropies as a consequence of a stabilizer symmetry, placing them among the Clifford-stabilizer states covered by the paper’s main theorem.

Gap

Issue #14173 records the literature check before any Lean: two arXiv versions; the only citing work found (arXiv:2607.07197) does not discuss the conjecture; arXiv, web and Zenodo searches found no settlement. not-found-in-searched-scope.

Route

The eigenphase-extended Clifford group ${\mathcal{C}}_{1,3}^{\prime }$ is finite, so there are only finitely many Clifford-stabilizer states modulo phase. The qutrit fiducials ${\psi }_z=(0,1,-z)/\sqrt{2}$, $|z|=1$, form a continuum of pairwise non-proportional states. This is the classical continuous family of qutrit SIC fiducials ([Tabia and Appleby 2013](../Library/QuantumStates/tabiaappleby2013qutrit.md) state it for $z=e^{2it}$, $t\in [0,\pi /6]$); the module proves the SIC condition for every unit $z$ directly. Hence some fiducial is not a Clifford-stabilizer state.

Falsifier

The refutation is for the paper’s definitions (single qudit, the eigenphase-extended Clifford group, uniqueness of the stabilized line); a weaker reading, for example “an eigenstate of some Clifford unitary”, is not addressed by result.

Evidence

The canonical source is D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.lean, with public zeta, pauliGroup, cliffordGroup, specialClifford, Lambda, eigenphaseClifford, invariantSubspace, IsNormalized, IsCliffordStabilizerState, IsSICFiducial, qutritFiducial, claim and result : ¬ claim. It reuses the frozen Weyl words WeylDisplacement.displacement and the frozen commutant theorem WindowRegister.window_commutant_eq_scalars (the clock and shift commute only with scalars). The axiom closure of result is exactly propext, Classical.choice and Quot.sound; there is no sorry, native_decide, or new axiom. The module statement is sha256:97d15233dd8ee9b1ac6ddc959dc2a90dc9177583ce8197b03aa278178cd50c78, the result statement sha256:c45352b51f54bba6cd4e2556959b8debb0b0fc32d95f87ee5295dedd4acb7624 and the claim statement sha256:fc36430abb9941ca4716ffe380ddde4ac06a4d1f7a5f5db9cdfb41e74403d232. The Freeze event is sha256:5dea10ff83cd1d3917616e488df413626188ddb09e2c0f65ffd46b8933d33fc2; its project-level prerequisite is the frozen WeylDisplacement module.

Triage

Tier 1 conjecture of a December 2025 paper, preregistered in issue #14173 before any Lean. theorem; resolution refuted.

The other theorems of the module (finiteness of the special and eigenphase-extended Clifford groups and of the stabilizer projectors at $d=3$, the fiducial property and injectivity of the qutrit family) lie on the proof path of result (CLAUDE.md §3.2 「有消费的辅助声明」). Utility is kind=certified-instance; basis=refutes (the claim and its refutation). There is no digestion atom.

What the refutation shows

Proved by result: the universal claim fails at $d=3$: there is a normalized qutrit SIC fiducial that spans no one-dimensional invariant subspace of any subset of the eigenphase-extended Clifford group ${\mathcal{C}}_{1,3}^{\prime }$.

Established inside the proof, for general data. ${\tilde{\mathcal{C}}}_{1,3}$, $\Lambda$ and ${\mathcal{C}}_{1,3}^{\prime }$ are finite; hence the set of rank-one projectors of Clifford-stabilizer states is finite. Every ${\psi }_z=(0,1,-z)/\sqrt{2}$ with $|z|=1$ is a SIC fiducial, and $z\mapsto {\psi }_z{\psi }_z^{†}$ is injective.

Argued and checked numerically, not formalized.

• Mechanism. For three sampled values of $z$, a numerical enumeration of the 216-element projective Clifford group finds exactly three elements fixing the ray of ${\psi }_z$, and after rescaling to fix ${\psi }_z$ their common invariant space is two-dimensional; for those samples the fiducial is an eigenvector of Clifford elements but not the unique invariant line. That the same holds for every generic $z$ is ASSUMED-UNVERIFIED. Independently of the samples, the fiducials that are Clifford-stabilizer states form a finite set of rays inside the continuous family (the finiteness proved inside result).
• What survives. The argument says nothing about dimensions in which the fiducials are isolated (finitely many up to Clifford equivalence); there the conjecture may still hold, and the weaker statement “every SIC fiducial is an eigenvector of some Clifford unitary” is not addressed.

Open. Whether every SIC fiducial in a prime dimension $d\ne 3$ is a Clifford-stabilizer state, and which members of the qutrit family are, are not settled here.

Effect on the paper. Conjecture 1 is false as stated; the explanation it proposes for the extremality of SIC fiducials does not cover the continuous qutrit family.

ASSUMED-UNVERIFIED

The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent proof.