A certified lower bound on the quantum-capacity threshold of the depolarizing channel

arXiv:2608.15870v1 Announce Type: new Abstract: The noise threshold below which the qubit depolarizing channel retains positive quantum capacity has been studied since 1996, and every improvement to date -- most recently the record of Agarwal et al. -- rests on floating-point evaluation of the coherent information. We give the first proven positivity results in this regime: an explicit 45-copy rank-two input state A, published as a 1,472-byte witness, with certified $I_c>0$ at the exact rational $p=16239/250000=0.064956$ (per-Pauli convention; total error rate $3p=0.194868$), beyond the best previously reported numerical value. Because the depolarizing family is a semigroup under composition, fixed-input coherent information is nonincreasing in $p$ on $[0,1/4]$, so a single certified point extends to the entire interval below it. The same machinery confines the positivity boundaries of A and of the strongest public state of Agarwal et al. to disjoint rational intervals separated by more than $1/25000$ -- to our knowledge the first proven ordering of two competing code states. Every computational claim in the paper reduces to a finite list of big-integer comparisons, checkable by a dependency-free few-hundred-line verifier whose soundness rests on three self-contained half-page lemmas. Payloads, certificates, and verifier accompany the paper as a supplementary artifact.

Abstract:The noise threshold below which the qubit depolarizing channel retains positive quantum capacity has been studied since 1996, and every improvement to date -- most recently the record of Agarwal et al. -- rests on floating-point evaluation of the coherent information. We give the first proven positivity results in this regime: an explicit 45-copy rank-two input state A, published as a 1,472-byte witness, with certified $I_c>0$ at the exact rational $p=16239/250000=0.064956$ (per-Pauli convention; total error rate $3p=0.194868$), beyond the best previously reported numerical value. Because the depolarizing family is a semigroup under composition, fixed-input coherent information is nonincreasing in $p$ on $[0,1/4]$, so a single certified point extends to the entire interval below it. The same machinery confines the positivity boundaries of A and of the strongest public state of Agarwal et al. to disjoint rational intervals separated by more than $1/25000$ -- to our knowledge the first proven ordering of two competing code states. Every computational claim in the paper reduces to a finite list of big-integer comparisons, checkable by a dependency-free few-hundred-line verifier whose soundness rests on three self-contained half-page lemmas. Payloads, certificates, and verifier accompany the paper as a supplementary artifact.

11 pages, 2 tables. Verification artifact: DOI https://doi.org/10.5281/zenodo.21962925

arXiv:2608.15870 [quant-ph]

(or arXiv:2608.15870v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.15870 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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A certified lower bound on the quantum-capacity threshold of the depolarizing channel

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