Exact optimality of the constructed rank-one encoder at fixed noise strength

Establish whether the rank-one encoder paired with the partial-trace decoder in the constructed noise family is globally optimal for every fixed noise parameter $p>0$.

Background

The paper constructs a feasible rank-one encoder and shows that, as p→0+p\to0^+, it attains the globally optimized quadratic coefficient of the rank-one gap. This establishes asymptotic optimality through quadratic order, but the argument does not prove that the same encoder–decoder pair is exactly optimal at any fixed positive value of pp. Numerical optimization supports the result but does not supply an analytic global certificate.

References

Higher order corrections to the rank-one optimum remain undetermined, so no closed form for $(\rho,N_p)$ or $(\rho,N_p)$ is obtained. Exact optimality of this pair at fixed $p>0$ remains open.

— High-Rank Encoding Can Improve Approximate Quantum Error Correction  (2609.00778 - Li et al., 1 Sep 2026) in Supplemental Material, Section S3, subsection “The exact quadratic gap coefficient”