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

At dimension five the full and non-symmetric aggregates of a two-qudit state fall below their conjectured lower bounds

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 $\Sigma (d)$ is the set of stochastic Lagrangian subspaces of ${\mathbb{F}}_d^6$, ${\mathcal{T}}_{\mathrm{sym}}=\{{\mathcal{T}}_O\mid O\in S_3\}$, ${\mathcal{T}}_{\mathrm{ns}}=\Sigma (d)\setminus {\mathcal{T}}_{\mathrm{sym}}$ and $\kappa (\Psi ,\mathcal{T})={\sum }_{\mathcal{T}\in \mathcal{T}}\kappa (\Psi ,\mathcal{T})$. The verbatim statements are in the literature note. The pointwise clause and the clause $\kappa (\Psi ,{\mathcal{T}}_{\mathrm{iso}})\ge 6$ are refuted ([pointwise](zhu-2024-clifford-third-moment-kappa-negativity.md), isotropic aggregate). Issue #14620 targets the two remaining lower bounds, $6\le \kappa (\Psi ,\Sigma (d))$ and $0\le \kappa (\Psi ,{\mathcal{T}}_{\mathrm{ns}})$.

Motivation

The paper uses the aggregate sums over $\Sigma (d)$ for the shadow norms of stabilizer projectors and for the third frame potential of Clifford orbits, and derives the aggregate bounds from the pointwise clause. Whether the lower bounds over $\Sigma (d)$ and ${\mathcal{T}}_{\mathrm{ns}}$ hold decides whether the consequences stated with them survive the failure of the pointwise clause.

Gap

The paper proves $4(D+d)/(D+1)\le \kappa (\Psi ,\Sigma (d))$ with $D=d^n$, which is at least $6$ only for one qudit, the upper bounds, and the conjecture for $d=3$. It states $|\Sigma (d)|=2d+2$ and, for $d\equiv 2(\mathrm{mod}3)$, $\Sigma (d)={\mathcal{T}}_{\mathrm{iso}}$. Issues #14561 and #14620 record the literature checks before any Lean: no later work proves or refutes the aggregate lower bounds; not-found-in-searched-scope.

Route

Let $\mathcal{T}\in \Sigma (5)$ and $(\mathbf{y};0)\in \mathcal{T}$. The quadratic condition gives $\mathbf{y}\cdot \mathbf{y}=0$, and applied to $(1+\mathbf{y};1)$ it gives ${\sum }_k{y}_k=0$. Then $\mathbf{y}\cdot \mathbf{y}=2(y_1^2+y_1{y}_2+y_2^2)$ with $-3$ a non-square mod $5$, so $\mathbf{y}=0$. The second projection is injective on the three-dimensional $\mathcal{T}$, so $\mathcal{T}={\mathcal{T}}_O$, and the conditions make $O$ a stochastic isometry. Hence $\Sigma (5)={\mathcal{T}}_{\mathrm{iso}}$. For the two-qudit state $\Psi =v/\sqrt{458}$ of #14561, $\kappa (\Psi ,\Sigma (5))=140241723/24017978<6$; each of the six permutation graphs has $\kappa =1$, so $\kappa (\Psi ,{\mathcal{T}}_{\mathrm{ns}})=-3866145/24017978<0$.

Falsifier

An error in the classification $\Sigma (5)={\mathcal{T}}_{\mathrm{iso}}$, in the identification of ${\mathcal{T}}_{\mathrm{sym}}$ or in the frozen two-qudit values. Exhaustive enumeration gives $|\Sigma (5)|=12$ with every element a graph, as the paper’s $|\Sigma (d)|=2d+2$ requires.

Evidence

The canonical source is D5/S3/Quantum/Magic/CliffordThirdMomentSigmaRefutation.lean, with public claim (the disjunction of the two lower bounds) 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 sha256:5e69691622247a60b0dfb251155c7bb412574c1cc80d163011458f6d54241347, the result statement sha256:52e7406bf235904808230d3d5fff4d7ac140f9f5b981894bd6f29778a06569a9, the claim statement sha256:969abd3b9602def77f7e1d6cd9ab6c13cc157b41fed903c3260da2c5616ad9e4 and the sigmaSubspaces_five statement sha256:e3df810a9f40214791e9c8732254144d2673d6f3c4fedfadfccb5ac5b40c2618. The Freeze event is sha256:c61cb9bb0f116f6787b705cd355848bd4436e40572e8886a161ff79dd10358b2; its one project-level prerequisite is the frozen node sha256:0630d507568cef406b80d8acc31aaccdd8f6709e5874320aff894907aa780a56 of D5/S3/Quantum/Magic/CliffordThirdMomentAggregateRefutation, whose graph subspaces, $O_3(d)$, ${\mathcal{T}}_{\mathrm{iso}}$, state and values the module reuses; IsStochasticLagrangian and kappa come from D5/S3/Quantum/Magic/CliffordThirdMomentNegativity.

Triage

Tier 1 printed clauses of an October 2024 conjecture, preregistered in issue #14620 before any Lean. theorem; resolution refuted.

Content declarations, each on the live proof of result: the public classification sigmaSubspaces_five ($\Sigma (5)={\mathcal{T}}_{\mathrm{iso}}$), the private kernel fact that no nonzero $\mathbf{y}\in {\mathbb{F}}_5^3$ has $\mathbf{y}\cdot \mathbf{y}=0={\sum }_k{y}_k$, the two lemmas that turn it into the graph description, and the values kappa_sigma_five and kappa_ns_five. The bind-only private lemma graph_stochastic (every ${\mathcal{T}}_O$ with $O\in O_3(d)$ satisfies the three conditions) is used by the classification and by kappa_ns_five. 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: both lower bounds $6\le \kappa (\Psi ,\Sigma (d))$ and $0\le \kappa (\Psi ,{\mathcal{T}}_{\mathrm{ns}})$ of Conjecture 2 fail for a normalized two-qudit state at $d=5$. Together with the earlier refutations, every lower bound of Conjecture 2 fails; the upper bounds are the paper’s theorems.

Argued, not formalized.

• 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.
• Primes $d\equiv 2(\mathrm{mod}3)$. The argument for $\Sigma (5)={\mathcal{T}}_{\mathrm{iso}}$ uses only that $-3$ is a non-square mod $d$, which holds exactly for $d\equiv 2(\mathrm{mod}3)$; it agrees with the paper’s statement for those primes.

Open. Whether the lower bounds over $\Sigma (d)$ hold for two qudits at $d\equiv 1(\mathrm{mod}3)$, where $\Sigma (d)$ contains four defect subspaces besides ${\mathcal{T}}_{\mathrm{iso}}$; the exact minimum of $\kappa (\Psi ,\Sigma (d))$ for $n\ge 2$.

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 and the statement $\Sigma (d)={\mathcal{T}}_{\mathrm{iso}}$ for $d\equiv 2(\mathrm{mod}3)$ unchanged is not verified firsthand. The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent refutation.