SIC fiducials beyond Clifford-stabilizer states

Abstract

The special Clifford group of a qutrit is finite, and its eigenphase extension has only finitely many one-dimensional fixed spaces. The continuous qutrit SIC family supplies infinitely many distinct pure-state projectors. Consequently, some SIC fiducials are not Clifford-stabilizer states.

Definition 1.1 (The half-period phase).

$$\forall d\in \mathbb{N},\mathrm{zeta}\left(d\right)=\exp \left(\frac{(\pi )\cdot (i)}{d}\right)$$

Formalization. D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.zeta (✓ std3).

Citation. Muhammad Erew and Moshe Goldstein (2025). Extremizing Measures of Magic on Pure States by Clifford-stabilizer States. DOI: 10.48550/arXiv.2512.19657.

Commentary.

The computational basis is indexed by ZMod d. The clock phase is exp(2 pi i / d), and zeta is exp(pi i / d). The shift sends basis vector j to basis vector j + 1. Write D(a,b) for the displacement X to the power a times Z to the power b.

Definition 1.2 (The single-qudit Pauli group).

$$\forall d\in \mathbb{N},(d\ne 0)\Rightarrow (\forall P\in \mathrm{Matrix}\left(\mathrm{ZMod}\left(d\right),\mathrm{ZMod}\left(d\right),\mathbb{C}\right),P\in \mathrm{pauliGroup}\left(d\right)\iff (\exists x\in \mathrm{ZMod}\left(2\right),\exists k\in \mathrm{ZMod}\left(d\right),\exists a\in \mathrm{ZMod}\left(d\right),\exists b\in \mathrm{ZMod}\left(d\right),P=(((-(1){)}^{\mathrm{val}\left(x\right)})\cdot ((\mathrm{zeta}\left(d\right){)}^{\mathrm{val}\left(k\right)}))\cdot (\mathrm{displacement}\left(d,a,b\right))))$$

Formalization. D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.pauliGroup (✓ std3).

Citation. Muhammad Erew and Moshe Goldstein (2025). Extremizing Measures of Magic on Pure States by Clifford-stabilizer States. DOI: 10.48550/arXiv.2512.19657.

Commentary.

The finite Pauli group includes the sign exponent x in ZMod 2 and the phase, shift and clock exponents k, a and b in ZMod d. Powers use the least nonnegative representatives of residues.

Definition 1.3 (The Clifford group).

$$\forall d\in \mathbb{N},(d\ne 0)\Rightarrow (\forall U\in \mathrm{Matrix}\left(\mathrm{ZMod}\left(d\right),\mathrm{ZMod}\left(d\right),\mathbb{C}\right),U\in \mathrm{cliffordGroup}\left(d\right)\iff ((U\in \mathrm{Unitary}\left(d\right))\wedge (\mathrm{image}\left(\mathrm{Ad}\left(U\right),\mathrm{pauliGroup}\left(d\right)\right)=\mathrm{pauliGroup}\left(d\right))))$$

Formalization. D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.cliffordGroup (✓ std3).

Citation. Muhammad Erew and Moshe Goldstein (2025). Extremizing Measures of Magic on Pure States by Clifford-stabilizer States. DOI: 10.48550/arXiv.2512.19657.

Commentary.

A Clifford matrix is unitary, and conjugation by it maps the entire Pauli group onto itself. Here Ad(U) sends P to U P U adjoint.

Definition 1.4 (The special Clifford group).

$$\forall d\in \mathbb{N},(d\ne 0)\Rightarrow (\forall U\in \mathrm{Matrix}\left(\mathrm{ZMod}\left(d\right),\mathrm{ZMod}\left(d\right),\mathbb{C}\right),U\in \mathrm{specialClifford}\left(d\right)\iff ((U\in \mathrm{cliffordGroup}\left(d\right))\wedge (\det \left(U\right)=1)))$$

Formalization. D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.specialClifford (✓ std3).

Citation. Muhammad Erew and Moshe Goldstein (2025). Extremizing Measures of Magic on Pure States by Clifford-stabilizer States. DOI: 10.48550/arXiv.2512.19657.

Commentary.

Special Clifford matrices have determinant one.

Definition 1.5 (The eigenvalue set).

$$\forall d\in \mathbb{N},(d\ne 0)\Rightarrow (\forall mu\in \mathbb{C},mu\in \mathrm{Lambda}\left(d\right)\iff (\exists U\in \mathrm{Matrix}\left(\mathrm{ZMod}\left(d\right),\mathrm{ZMod}\left(d\right),\mathbb{C}\right),(U\in \mathrm{specialClifford}\left(d\right))\wedge (\exists v\in (\mathrm{ZMod}\left(d\right)\to \mathbb{C}),(v\ne 0)\wedge ((U)\cdot (v)=(mu)\cdot (v)))))$$

Formalization. D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.Lambda (✓ std3).

Citation. Muhammad Erew and Moshe Goldstein (2025). Extremizing Measures of Magic on Pure States by Clifford-stabilizer States. DOI: 10.48550/arXiv.2512.19657.

Commentary.

Lambda contains every eigenvalue of every special Clifford matrix. An eigenvalue has a nonzero eigenvector.

Definition 1.6 (The eigenphase extension).

$$\forall d\in \mathbb{N},(d\ne 0)\Rightarrow (\forall V\in \mathrm{Matrix}\left(\mathrm{ZMod}\left(d\right),\mathrm{ZMod}\left(d\right),\mathbb{C}\right),V\in \mathrm{eigenphaseClifford}\left(d\right)\iff (\exists mu\in \mathbb{C},(mu\in \mathrm{Lambda}\left(d\right))\wedge (\exists U\in \mathrm{Matrix}\left(\mathrm{ZMod}\left(d\right),\mathrm{ZMod}\left(d\right),\mathbb{C}\right),(U\in \mathrm{specialClifford}\left(d\right))\wedge (V=(mu)\cdot (U)))))$$

Formalization. D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.eigenphaseClifford (✓ std3).

Citation. Muhammad Erew and Moshe Goldstein (2025). Extremizing Measures of Magic on Pure States by Clifford-stabilizer States. DOI: 10.48550/arXiv.2512.19657.

Commentary.

The extension consists of all scalar multiples mu U with U special Clifford and mu in Lambda.

Definition 1.7 (The pointwise fixed space).

$$\forall d\in \mathbb{N},(d\ne 0)\Rightarrow (\forall S\in \mathrm{Set}\left(\mathrm{Matrix}\left(\mathrm{ZMod}\left(d\right),\mathrm{ZMod}\left(d\right),\mathbb{C}\right)\right),\forall v\in (\mathrm{ZMod}\left(d\right)\to \mathbb{C}),v\in \mathrm{invariantSubspace}\left(S\right)\iff (\forall U\in \mathrm{Matrix}\left(\mathrm{ZMod}\left(d\right),\mathrm{ZMod}\left(d\right),\mathbb{C}\right),(U\in S)\Rightarrow ((U)\cdot (v)=v)))$$

Formalization. D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.invariantSubspace (✓ std3).

Citation. Muhammad Erew and Moshe Goldstein (2025). Extremizing Measures of Magic on Pure States by Clifford-stabilizer States. DOI: 10.48550/arXiv.2512.19657.

Commentary.

The invariant subspace is the intersection of the kernels of U minus the identity over U in S. The condition is pointwise fixation, U v = v.

Definition 1.8 (Pure-state normalization).

$$\forall d\in \mathbb{N},(d\ne 0)\Rightarrow (\forall psi\in (\mathrm{ZMod}\left(d\right)\to \mathbb{C}),\mathrm{IsNormalized}\left(psi\right)\iff (\mathrm{inner}\left(psi,psi\right)=1))$$

Formalization. D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.IsNormalized (✓ std3).

Citation. Muhammad Erew and Moshe Goldstein (2025). Extremizing Measures of Magic on Pure States by Clifford-stabilizer States. DOI: 10.48550/arXiv.2512.19657.

Commentary.

The Hermitian inner product is the sum of conjugate(psi j) times phi j. Normalization is inner(psi,psi) = 1, equivalently Hilbert norm one.

Definition 1.9 (Clifford-stabilizer states).

$$\forall d\in \mathbb{N},(d\ne 0)\Rightarrow (\forall psi\in (\mathrm{ZMod}\left(d\right)\to \mathbb{C}),\mathrm{IsCliffordStabilizerState}\left(d,psi\right)\iff ((\mathrm{IsNormalized}\left(psi\right))\wedge (\exists S\in \mathrm{Set}\left(\mathrm{Matrix}\left(\mathrm{ZMod}\left(d\right),\mathrm{ZMod}\left(d\right),\mathbb{C}\right)\right),(S\subseteq \mathrm{eigenphaseClifford}\left(d\right))\wedge (\mathrm{invariantSubspace}\left(S\right)=\mathrm{span}\left(\mathbb{C},\left\{psi\right\}\right)))))$$

Formalization. D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.IsCliffordStabilizerState (✓ std3).

Citation. Muhammad Erew and Moshe Goldstein (2025). Extremizing Measures of Magic on Pure States by Clifford-stabilizer States. DOI: 10.48550/arXiv.2512.19657.

Commentary.

A normalized state is Clifford-stabilized when its complex line is exactly the pointwise fixed space of a subset of the eigenphase extension. This is the fixed-space formulation of the source’s convention.

Definition 1.10 (SIC fiducials).

$$\forall d\in \mathbb{N},(d\ne 0)\Rightarrow (\forall psi\in (\mathrm{ZMod}\left(d\right)\to \mathbb{C}),\mathrm{IsSICFiducial}\left(d,psi\right)\iff ((\mathrm{IsNormalized}\left(psi\right))\wedge (\forall a\in \mathrm{ZMod}\left(d\right),\forall b\in \mathrm{ZMod}\left(d\right),((a,b)\ne (0,0))\Rightarrow (\mathrm{norm}\left(\mathrm{inner}\left(psi,(\mathrm{displacement}\left(d,a,b\right))\cdot (psi)\right)\right)=\frac{1}{\mathrm{sqrt}\left((d)+(1)\right)}))))$$

Formalization. D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.IsSICFiducial (✓ std3).

Citation. Muhammad Erew and Moshe Goldstein (2025). Extremizing Measures of Magic on Pure States by Clifford-stabilizer States. DOI: 10.48550/arXiv.2512.19657.

Commentary.

Every nonidentity displacement has overlap modulus one divided by the square root of d + 1. Multiplying a displacement by a unit-modulus phase preserves this condition.

Definition 1.11 (The proposed stabilizer nature of SIC fiducials).

$$claim\iff (\forall d\in \mathbb{N},(\mathrm{Prime}\left(d\right))\Rightarrow (\forall psi\in (\mathrm{ZMod}\left(d\right)\to \mathbb{C}),(\mathrm{IsSICFiducial}\left(d,psi\right))\Rightarrow (\mathrm{IsCliffordStabilizerState}\left(d,psi\right))))$$

Formalization. D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.claim (✓ std3).

Citation. Muhammad Erew and Moshe Goldstein (2025). Extremizing Measures of Magic on Pure States by Clifford-stabilizer States. DOI: 10.48550/arXiv.2512.19657.

Commentary.

The conjecture quantifies over all prime dimensions and all single-qudit SIC fiducials.

Theorem 1.12 (Finiteness of the special qutrit Clifford group).

$$\mathrm{Finite}\left(\mathrm{specialClifford}\left(3\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.specialClifford_three_finite (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Muhammad Erew and Moshe Goldstein (2025). Extremizing Measures of Magic on Pure States by Clifford-stabilizer States. DOI: 10.48550/arXiv.2512.19657.

Commentary.

Conjugating X and Z gives a pair of Pauli matrices, so there are finitely many such pairs. If U and V give the same pair, V adjoint times U commutes with both X and Z. The clock-and-shift commutant consists of scalar matrices. Determinant one forces that scalar to be a cube root of unity. Thus every fiber contains finitely many matrices.

Theorem 1.13 (Finiteness of the eigenphase extension).

$$\mathrm{Finite}\left(\mathrm{eigenphaseClifford}\left(3\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.eigenphaseClifford_three_finite (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Muhammad Erew and Moshe Goldstein (2025). Extremizing Measures of Magic on Pure States by Clifford-stabilizer States. DOI: 10.48550/arXiv.2512.19657.

Commentary.

A finite-dimensional matrix has finitely many eigenvalues. Taking the union over the finite special Clifford group leaves Lambda finite, and taking scalar multiples from two finite sets leaves the extension finite.

Theorem 1.14 (Finitely many Clifford-stabilizer projectors).

$$\mathrm{Finite}\left(\{P\mid \exists psi\in (\mathrm{ZMod}\left(3\right)\to \mathbb{C}),(\mathrm{IsCliffordStabilizerState}\left(3,psi\right))\wedge (P=(psi)\cdot (\mathrm{adjoint}\left(psi\right)))\}\right)$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.stabilizer_projectors_three_finite (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Muhammad Erew and Moshe Goldstein (2025). Extremizing Measures of Magic on Pure States by Clifford-stabilizer States. DOI: 10.48550/arXiv.2512.19657.

Commentary.

There are finitely many subsets of the eigenphase extension. Each subset determines one fixed space. Two normalized vectors spanning the same complex line have the same rank-one projector, so the set of resulting projectors is finite.

Definition 1.15 (The continuous qutrit family).

$$\forall z\in \mathbb{C},\mathrm{qutritFiducial}\left(z\right)=\frac{(0,1,-(z))}{\mathrm{sqrt}\left(2\right)}$$

Formalization. D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.qutritFiducial (✓ std3).

Citation. Gelo Noel M. Tabia; D. M. Appleby (2013). Exploring the geometry of qutrit state space using symmetric informationally complete probabilities. DOI: 10.1103/PhysRevA.88.012131. URL: https://arxiv.org/abs/1304.8075v2.

Commentary.

For each complex parameter z, the column vector has amplitudes 0, 1 and -z, divided by the square root of two.

Theorem 1.16 (A unit circle of SIC fiducials).

$$\forall z\in \mathbb{C},(\mathrm{norm}\left(z\right)=1)\Rightarrow (\mathrm{IsSICFiducial}\left(3,\mathrm{qutritFiducial}\left(z\right)\right))$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.qutritFiducial_isSIC (✓ std3). ∎

Citation. Gelo Noel M. Tabia; D. M. Appleby (2013). Exploring the geometry of qutrit state space using symmetric informationally complete probabilities. DOI: 10.1103/PhysRevA.88.012131. URL: https://arxiv.org/abs/1304.8075v2.

Commentary.

For modulus-one z the vector is normalized. When a = 0 and b is nonzero, the overlap is -1/2 because 1 + omega + omega squared = 0. When a = 1 the overlap is -conjugate(z) omega to the power b divided by two; when a = 2 it is -z omega to the power 2b divided by two. Each has modulus 1/2, equal to one divided by the square root of 3 + 1. Tabia and Appleby state this family for z = e^{2it} with t in [0, pi/6]; the proof here covers every unit z by direct computation.

Theorem 1.17 (Distinct parameters give distinct projectors).

$$\forall z\in \mathbb{C},\forall w\in \mathbb{C},((\mathrm{qutritFiducial}\left(z\right))\cdot (\mathrm{adjoint}\left(\mathrm{qutritFiducial}\left(z\right)\right))=(\mathrm{qutritFiducial}\left(w\right))\cdot (\mathrm{adjoint}\left(\mathrm{qutritFiducial}\left(w\right)\right)))\Rightarrow (z=w)$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.qutrit_projector_injective (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Muhammad Erew and Moshe Goldstein (2025). Extremizing Measures of Magic on Pure States by Clifford-stabilizer States. DOI: 10.48550/arXiv.2512.19657.

Commentary.

The projector entry in row 1, column 2 is -conjugate(z)/2. Equality of projectors therefore forces equality of the parameters.

Theorem 1.18 (Refutation in dimension three).

$$\neg (\forall d\in \mathbb{N},(\mathrm{Prime}\left(d\right))\Rightarrow (\forall psi\in (\mathrm{ZMod}\left(d\right)\to \mathbb{C}),(\mathrm{IsSICFiducial}\left(d,psi\right))\Rightarrow (\mathrm{IsCliffordStabilizerState}\left(d,psi\right))))$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.result (✓ std3). ∎

Resolves. Problems/erew-goldstein-2025-sic-clifford-stabilizer (refuted) by D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.result.

Source. Repository-derived.

Acknowledgement. Muhammad Erew and Moshe Goldstein (2025). Extremizing Measures of Magic on Pure States by Clifford-stabilizer States. DOI: 10.48550/arXiv.2512.19657.

Commentary.

The complex unit circle is infinite: its real-coordinate image contains the entire range of cosine. The qutrit family maps that circle injectively to pure-state projectors. If every SIC fiducial were Clifford-stabilized, this infinite image would be a subset of the finite set of Clifford-stabilizer projectors. Hence the conjecture is false, already in prime dimension three.

References

• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.IsCliffordStabilizerState
• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.IsNormalized
• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.IsSICFiducial
• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.Lambda
• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.claim
• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.cliffordGroup
• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.eigenphaseClifford
• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.eigenphaseClifford_three_finite
• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.invariantSubspace
• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.pauliGroup
• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.qutritFiducial
• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.qutritFiducial_isSIC
• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.qutrit_projector_injective
• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.result
• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.specialClifford
• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.specialClifford_three_finite
• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.stabilizer_projectors_three_finite
• Truth anchor: D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.zeta
• Dependency: D5/S3/Quantum/Algebra/WeylDisplacement