The Quantum Channel That Won't Give Up express
In 1996, physicists asked: how much noise can a quantum channel withstand before it stops transmitting information? A phone call where every third word is replaced by random noise is still understandable. Up to what level of interference can the listener still grasp the meaning? For quantum channels, this threshold has been sought for nearly thirty years, and all previous estimates relied on floating-point computer calculations: that is, on approximations.
This new work provides the first rigorous proof. The authors constructed an explicit state of 45 qubits (quantum analogs of bits) that retains the ability to transmit information at an error rate of 6.5% per qubit. This is worse than previously thought, but now it is not an estimate but a proven fact. The entire computation reduces to a finite list of comparisons of large integers, and anyone can verify it with a short program.
The most surprising part: the authors didn't just find a working state; they also strictly ordered two competing states that were previously only compared numerically. It turns out their survival boundaries are separated by an interval of more than 1/25000—like two runners finishing within a thousandth of a second, and now we have a photo finish.
The practical benefit lies in trust. Quantum computing and communication require precise guarantees, not 'we checked on a computer, seems to work.' Now, for one important class of channels, such guarantees exist and are reproducible: all data and programs are publicly available.
Original abstract
as written by the authors arXiv:2608.15870v1
The noise threshold below which the qubit depolarizing channel retains positive quantum capacity has been studied since 1996. The classic constructions come with exact finite formulae, but in every reported threshold to date -- most recently the record of Agarwal et al. -- the final step, the sign of the coherent information at the reported point, is a floating-point evaluation. We give the first exact-arithmetic, independently machine-checkable positivity proofs 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 both the best value printed in the Agarwal et al. paper ($$0.064657$$) and the strongest state in the same authors' public repository ($$\approx 0.064911$$), both numerical. 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 the positivity boundaries of two explicit 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.
Verifying all the calculations in the paper takes less than a minute on a regular laptop, although the calculations themselves required months of work.