bibkey: zhu2024thirdmoments authors: H. Zhu, C. Mao, C. Yi year: 2024 title: “Third moments of qudit Clifford orbits and 3-designs based on magic orbits” doi: 10.48550/arXiv.2410.13575 url: https://arxiv.org/abs/2410.13575v1 claim: “Conjecture 2: for an odd prime d and |Ψ⟩ ∈ H_d^{⊗n}, 0 ≤ κ(Ψ,𝒯) ≤ 1 for every stochastic Lagrangian subspace 𝒯 ∈ Σ(d), together with three aggregate inequalities that follow from it; κ(Ψ,𝒯) = tr[R(𝒯)(|Ψ⟩⟨Ψ|)^{⊗3}], R(𝒯) = r(𝒯)^{⊗n}, r(𝒯) = Σ_{(x;y)∈𝒯} |x⟩⟨y|.” strata_touched:

• D5/S3/Quantum/Magic/CliffordThirdMomentNegativity
• D5/S3/Quantum/Magic/CliffordThirdMomentAggregateRefutation
• D5/S3/Quantum/Magic/CliffordThirdMomentSigmaRefutation license: citation-only triage: anchor

Zhu, Mao and Yi, third moments of qudit Clifford orbits

H. Zhu, C. Mao and C. Yi, Third moments of qudit Clifford orbits and 3-designs based on magic orbits, arXiv:2410.13575v1 (17 October 2024; quant-ph, cross-listed math-ph); Commun. Math. Phys. 407, 242 (2026), DOI 10.1007/s00220-026-05583-8.

Verified locator

DOI: 10.48550/arXiv.2410.13575. Primary version: https://arxiv.org/abs/2410.13575v1 (the only arXiv version). The TeX source 3rdMoment.tex of v1 supplies the stochastic Lagrangian subspaces (sec:SLSspanningSet), the operators $r(\mathcal{T})$ and $R(\mathcal{T})$ (eq:rRT), the definition of $\kappa (\Psi ,\mathcal{T})$ (eq:kappapsiT), the stochastic orthogonal group (sec:Otd, eq:TOdef), the subsets of $\Sigma (d)$ (eq:scrTSIEO), the sums over subsets (lines 1945–1948) and Conjecture 2 (con:kappaTLUB, Eq. (147)).

Source statements

Stochastic Lagrangian subspaces: “A subspace \caT \leq \bbF_d^{2t} … is a \emph{stochastic Lagrangian subspace} if it satisfies the following three conditions: $\bfx \cdot \bfx - \bfy \cdot \bfy = 0$forany$(\bfx;\bfy) \in \caT$.$\caT$hasdimension$t$.$\mathbf{1}_{2t} \in \caT$.” $\Sigma (d):={\Sigma }_{3,3}(d)$.

Operators: “r(\caT):= \sum_{(\bfx;\bfy) \in \caT} |\bfx\> \<\bfy|, \quad R(\caT):= r(\caT)^{\otimes n}”.

Moments: “\kappa(\Psi,\caT):=\tr[R(\caT)(|\Psi\>\<\Psi|)^{\otimes 3}]”; for a subset, “\kappa(\Psi,\scrT,j):=\sum_{\caT\in \scrT}\kappa^j(\Psi,\caT) … abbreviated as \kappa(\Psi,\scrT) … when $j=1$”.

Stochastic orthogonal group: “A $t\times t$ matrix $O$ over \bbF_d is a stochastic isometry if it satisfies the following two conditions: O\bfx\cdot O\bfx=\bfx\cdot \bfx for any \bfx \in \bbF_d^t. $O\cdot 1_t=1_t$ … The two conditions imply that O^\top O=OO^\top=\mathds{1} … The stochastic orthogonal group $O_t(d)$ is the group of all stochastic isometries on \bbF_d^t”, with “\caT_O:=\left\{ (O\bfx;\bfx)\, | \, \bfx \in \bbF_d^t \right\}”.

Subsets of $\Sigma (d)$: “\scrT_\sym:=\{\caT_O\ |\ O\in S_3\}, \scrT_\ns:=\Sigma(d)\setminus\scrT_\sym, \scrT_\iso:= \{T_O \; | \; O \in O_3(d)\}”.

Cardinality (eq:Sigma33DefectNum): “|\Sigma(d)|=2d+2, \quad |\scrT_\defe|=|\Sigma(d)|-|O_3(d)|= \begin{cases} 2 & d=3,\\ 4 & d = 1\mmod 3,\\ 0 & d=2\mmod 3. \end{cases}” followed by “When d = 2\mmod 3, \scrT_\defe is empty and \Sigma(d)=\scrT_\iso contains only one double coset.”

Conjecture 2: “Suppose $d$ is an odd prime and |\Psi\>\in\caH_d^{\otimes n}. Then 0\leq \kappa(\Psi,\caT)\leq 1 \quad \forall \caT\in \Sigma(d), \quad 6\leq \kappa(\Psi,\Sigma(d))\leq 2d+2, \quad 0\leq \kappa(\Psi,\scrT_\ns)\leq 2d-4, \quad \kappa(\Psi, \scrT_\iso)\geq 6.” followed by “If the first inequality in \eref{eq:kappaTLUBcon} holds, then all inequalities hold thanks to \lref{lem:kappaTLUB}.”

Scope

The paper proves $-1\le \kappa (\Psi ,\mathcal{T})\le 1$ in general (Lemma 20), the conjecture for $d=3$, $\kappa =1$ for stabilizer states (Proposition 4) and nonnegativity for its cubic-phase magic-state families.