bibkey: lijiang2026highrank authors: Bikun Li; Liang Jiang year: 2026 title: “High-Rank Encoding Can Improve Approximate Quantum Error Correction” doi: null url: https://arxiv.org/abs/2609.00778v1 claim: “Exact optimality of this pair at fixed p > 0 remains open.” strata_touched:

• D5/S3/Quantum/Recovery/LiJiangPolarPairOptimalityRefutation license: citation-only triage: anchor

High-Rank Encoding Can Improve Approximate Quantum Error Correction

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• URL: https://arxiv.org/abs/2609.00778v1 (version 1, the source record inspected).
• Scope: printed p. 17 immediately after Eq. S64 for the fixed-noise optimality statement; Eqs. 18–23 and S60 on printed pp. 16–17 for the definitions used by the repository refutation.

Bikun Li and Liang Jiang, arXiv:2609.00778v1, 1 September 2026, quant-ph. The source is this version, rather than an unverified later or journal text. Printed p. 17, immediately after Eq. S64, states:

Exact optimality of this pair at fixed p > 0 remains open.

The pair is $({\mathcal{C}}_{V_p(\chi )},{\mathrm{tr}}_S)$, where Eq. S60 on printed p. 16 defines $C_p(\chi )=Q_p(I_d\otimes |\chi ⟩)$ and $V_p(\chi )=C_p(\chi )[C_p(\chi {)}^{†}{C}_p(\chi ){]}^{-1/2}$. The inverse square root is the positive Gram square root followed by its inverse. The code space is ${\mathbb{C}}^d\otimes {\mathbb{C}}^d$ in logical, auxiliary order; ${\mathrm{tr}}_S$ removes the right factor.

Eqs. 18–23 specify the actual noise. In the notation of the source module,

$${\lambda }_1=1-\frac{p}{d^2-2},{\lambda }_4=\frac{d^2-1+p}{d^2},{\lambda }_5=\frac{1-p}{d^2},{\lambda }_2=\frac{(1-{\lambda }_1)(d^2{\lambda }_1-1)}{d^2{\lambda }_4},{\lambda }_3=\frac{2(1-{\lambda }_1{)}^2}{d^2{\lambda }_4}.$$

For $\psi =d^{-1/2}{\sum }_i|ii⟩$ and $\Pi =|\psi ⟩⟨\psi |$, Eq. 18 is $Q_p=\sqrt{d}[\sqrt{{\lambda }_4/(d^2-1)}(I-\Pi )+\sqrt{{\lambda }_5}\Pi ]$. Eqs. 19 and 22 give $D_j=I\otimes ⟨j|$, $B_{ij}=Q{D}_i^{†}{D}_j$, and $E_{ij}=(|ij⟩⟨\psi |-\sqrt{{\lambda }_5}{B}_{ij}^{†})/\sqrt{{\lambda }_4}$. Eq. 23 acts on every code matrix $X$ by

$${\mathcal{N}}_p(X)={\lambda }_3\mathrm{tr}(X)I+({\lambda }_1-{\lambda }_3){\mathrm{tr}}_S(Q_{p}X{Q}_p)\otimes I+{\lambda }_2\sum _{i,j}{E}_{ij}X{E}_{ij}^{†}.$$

The optimization class in Eq. 13 comprises completely positive trace-nonincreasing encoders and decoders, with encoder Choi rank at most one for the rank-one optimum. The source uses input-first Choi order. For $\tau =I_d/d$, its canonical-purification fidelity is the unrenormalized overlap $⟨\psi |[(\mathcal{D}\circ {\mathcal{N}}_p\circ \mathcal{C})\otimes \mathrm{id}](|\psi ⟩⟨\psi |)|\psi ⟩$. No output trace normalization is applied. The source admits $0\le p<1$; the question at nonzero noise has $d\ge 2$ and $0<p<1$.

The module refutes the universal exact-optimality reading at $d=2$, $p=9/13$, $\chi =|0⟩$. It preserves the actual noise and decoder and replaces one polar column by a strictly better feasible column. It does not determine the global rank-one optimum. The source’s optimal quadratic asymptotic coefficient in Eqs. S63–S64 and its high-rank advantage theorem are not refuted.

The preregistration is issue #13324. Its bounded literature check of 5 October 2026 reports not-found-in-searched-scope. It inspected the single citing paper arXiv:2609.40203v1, whose single-channel convex iteration does not settle this joint pair question, and the unrelated author paper arXiv:2609.17439. Semantic Scholar was rate limited; nonindexed/private work and worldwide priority remain unverified. This note attributes the question and source definitions to Li and Jiang; the refutation is a repository result.