slug: zhu-2024-clifford-third-moment-aggregate-lower-bound
bibkey: zhu2024thirdmoments
doi: 10.48550/arXiv.2410.13575
url: https://arxiv.org/abs/2410.13575v1
triage: theorem
motivation_gids:
  - D5/S3/Quantum/Magic/CliffordThirdMomentAggregateRefutation.result

The isotropic aggregate of a two-qudit state in dimension five is below six

Problem

H. Zhu, C. Mao and C. Yi, Third moments of qudit Clifford orbits and 3-designs based on magic orbits, arXiv:2410.13575v1, Section VII.1, Conjecture 2, asserts for an odd prime $d$ and $|\Psi ⟩\in {\mathcal{H}}_d^{\otimes n}$ the pointwise bounds $0\le \kappa (\Psi ,\mathcal{T})\le 1$ on $\Sigma (d)$ together with the aggregate bounds $6\le \kappa (\Psi ,\Sigma (d))\le 2d+2$, $0\le \kappa (\Psi ,{\mathcal{T}}_{\mathrm{ns}})\le 2d-4$ and $\kappa (\Psi ,{\mathcal{T}}_{\mathrm{iso}})\ge 6$. Here $\kappa (\Psi ,\mathcal{T})={\sum }_{\mathcal{T}\in \mathcal{T}}\kappa (\Psi ,\mathcal{T})$ and ${\mathcal{T}}_{\mathrm{iso}}=\{{\mathcal{T}}_O:O\in O_3(d)\}$. The verbatim statements are in the literature note.

The pointwise clause was refuted earlier with one qudit at $d=11$ ([dossier](zhu-2024-clifford-third-moment-kappa-negativity.md)); for that state the aggregate bounds hold. Issue #14561 targets the remaining aggregate lower bound in its printed form $\kappa (\Psi ,{\mathcal{T}}_{\mathrm{iso}})\ge 6$.

Motivation

The paper derives the aggregate bounds from the pointwise clause and uses the aggregate sums for the shadow norms of stabilizer projectors and for the third frame potential of Clifford orbits. With the pointwise clause refuted, whether the aggregate lower bounds still hold decides whether those consequences survive.

Gap

The paper proves $4(D+d)/(D+1)\le \kappa (\Psi ,\Sigma (d))$ with $D=d^n$, which is at least $6$ for one qudit, and the upper bounds; the conjecture for $d=3$; and positivity for its cubic-phase magic-state families. Issue #14561 records the literature check before any Lean: no later work proves or refutes the aggregate lower bounds; not-found-in-searched-scope.

Route

At $d=5$ the stochastic orthogonal group has twelve elements: the six permutation matrices and the six row orders of the rows $(3,4,4)$, $(4,3,4)$, $(4,4,3)$. For a graph subspace, $\kappa (\Psi ,{\mathcal{T}}_O)={\sum }_Y{\prod }_{k=1}^3\Psi (Y_k)\overline{\Psi ((OY{)}_k)}$. Reordering the rows of $O$ permutes the three copies and leaves $\kappa$ unchanged. For the two-qudit state $\Psi =v/\sqrt{458}$ of #14561, each permutation gives $\kappa =1$ and each other matrix gives $-2577430/458^3$, so $\kappa (\Psi ,{\mathcal{T}}_{\mathrm{iso}})=140241723/24017978<6$.

Falsifier

An error in the identification of $O_3(5)$ or in the integer sum, or an intended reading of ${\mathcal{T}}_{\mathrm{iso}}$ other than the graphs of $O_3(d)$. The positive controls $|00⟩$ and the uniform state give $\kappa (\Psi ,{\mathcal{T}}_{\mathrm{iso}})=12$, as the paper’s Proposition 4 requires for stabilizer states.

Evidence

The canonical source is D5/S3/Quantum/Magic/CliffordThirdMomentAggregateRefutation.lean, with public claim and result : ¬ claim. 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 sha256:87fa0d914d3520854d77069904b2c94a0b4e436b95cded2724dd4aede32fca66, the result statement sha256:ab48ed35cda00d5b5ba176f1008985182fada677985722d41f50b46958fef103 and the claim statement sha256:ab9e4fef8616bd26d3d8aadc3b24de1f1b45f45c9af27f30edc00c309fdcb1e5. The Freeze event is sha256:0630d507568cef406b80d8acc31aaccdd8f6709e5874320aff894907aa780a56; its one project-level prerequisite is the frozen node sha256:6f1e7f1909bb5bd57dcbacf1b57dd272d84520dc0cf284c3de6f6235b4209edd of D5/S3/Quantum/Magic/CliffordThirdMomentNegativity, whose R, stateCube and kappa the module reuses.

Triage

Tier 1 printed clause of an October 2024 conjecture (open after the pointwise refutation), preregistered in issue #14561 before any Lean. theorem; resolution refuted.

Content declarations, each on the live proof of result: group_five (the twelve-element description of $O_3(5)$, from a kernel classification of the stochastic rows), normalized_psi, the integer slices and integer_sum (the $15625$-term value $-2577430$), kappa_base and kappa_iso. Bind-only declarations, also on that proof: graphSubspace_injective, stochasticOrthogonal_five_iff, the graph collapse kappa_graph, the row-order invariance kappa_rows and kappa_identity (reindexing, instantiation and normalization). The settlement is admitted as an external open-problem resolution.

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 aggregate lower bound $\kappa (\Psi ,{\mathcal{T}}_{\mathrm{iso}})\ge 6$ of Conjecture 2 fails for a normalized two-qudit state at $d=5$.

Proved by D5/S3/Quantum/Magic/CliffordThirdMomentSigmaRefutation.result: the same state violates the lower bounds $6\le \kappa (\Psi ,\Sigma (d))$ and $0\le \kappa (\Psi ,{\mathcal{T}}_{\mathrm{ns}})$, through $\Sigma (5)={\mathcal{T}}_{\mathrm{iso}}$ ([dossier](zhu-2024-clifford-third-moment-sigma-lower-bounds.md)).

Argued, not formalized.

• The pointwise clause at $d=5$. Each non-permutation value is negative.
• Every $n\ge 2$. Graph expectations are multiplicative under tensor products and equal $1$ for a stabilizer state, so $\Psi \otimes |0{⟩}^{\otimes (n-2)}$ has the same values.
• Mechanism. The proven bound $4(D+d)/(D+1)$ is at least $6$ only for one qudit. Entanglement is necessary at $d=5$: for one qudit the six non-permutation values coincide (row-order invariance), say $a(\varphi )$, and the bound $8d/(d+1)=20/3$ gives $6+6a(\varphi )\ge 20/3$, so $a(\varphi )\ge 1/9$; for a product state $\varphi \otimes \chi$ graph expectations multiply, so $\kappa (\varphi \otimes \chi ,{\mathcal{T}}_{\mathrm{ns}})=6a(\varphi )a(\chi )\ge 2/27>0$. Numerical minimization found no violation for one qudit at $d=3,5,7$ or for two qudits at $d=3$; the smallest value of $\kappa (\Psi ,{\mathcal{T}}_{\mathrm{ns}})$ it found for two qudits at $d=5$ is $-1/6$, which is not certified as the global minimum.

Open. Whether the aggregate lower bounds hold for two qudits at $d=7$ and higher primes, the exact minimum of $\kappa (\Psi ,\Sigma (d))$ for $n\ge 2$, and how the paper’s conditional consequences for shadow norms change below $6$, are not determined here.

ASSUMED-UNVERIFIED

The main text of the published Communications in Mathematical Physics version is subscription-gated and was not read; that it prints Conjecture 2 unchanged is not verified firsthand. The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent refutation.