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The unassisted classical capacity of the generalized amplitude-damping channel

The generalized amplitude-damping channel Aγ,ν\mathcal A_{\gamma,\nu} is the qubit channel modelling energy relaxation with damping γ∈[0,1]\gamma\in[0,1] toward a thermal state with excited population ν∈[0,1]\nu\in[0,1]. By the Holevo-Schumacher-Westmoreland theorem its unassisted classical capacity is the regularized Holevo information lim⁡nχ(A⊗n)/n\lim_n\chi(\mathcal A^{\otimes n})/n, and since Holevo additivity fails in general (Hastings) the one-use value need not be the answer. Leditzky, Kaur, Datta and Wilde (2018) listed the classical capacity of amplitude damping as open, and Khatri, Sharma and Wilde (2020) gave only upper bounds for the generalized channel; exact values were known only where King's unital theorem or Shor's entanglement-breaking theorem applies. What is the classical capacity of Aγ,ν\mathcal A_{\gamma,\nu} for all γ,ν\gamma,\nu, and does it equal the one-shot Holevo capacity?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Quantum Shannon theory; channel capacities; Holevo additivity
Posed by
Leditzky, Kaur, Datta and Wilde, Phys. Rev. A 97 (2018) (amplitude damping open); Khatri, Sharma and Wilde, Phys. Rev. A 102 (2020) (only bounds for the generalized channel)
Year posed
2018
Years open
8y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
16 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims that for every γ,ν∈[0,1]\gamma,\nu\in[0,1] the unassisted classical capacity of Aγ,ν\mathcal A_{\gamma,\nu} equals its one-shot Holevo capacity, given by max⁡p∈[0,1][h((1−γ)p+γν)−g(γν(1−ν)+γ(1−γ)(p−ν)2)]\max_{p\in[0,1]}[h((1-\gamma)p+\gamma\nu)-g(\gamma\nu(1-\nu)+\gamma(1-\gamma)(p-\nu)^2)] in bits, with gg the entropy of a qubit state of given determinant, attained by two equiprobable pure signals of opposite phase and their independent products. It also claims additivity of one-shot Holevo capacity, minimum output entropy and regularized classical capacity under tensoring with any finite-dimensional channel. It does NOT give the quantum or private capacity of the channel, entanglement-assisted settings, or a closed form for the maximizing pp.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscript is credited to OpenAI with no human author named. The capacity theorem has a Lean formalization listed in the release's formalization catalogue.

Verification

No independent mathematician has checked this yet. The abstract and main theorem were read against the question. formalization.yaml lists ComparatorChallenges/AmplitudeDamping.json with declaration OAI.GAD.main in OAI/InformationTheory/AmplitudeDamping/Main.lean. The comparator statement, read here, says for all γ,ν∈[0,1]\gamma,\nu\in[0,1]: the n-use Holevo quantity is n times the one-use value, equals n times an attained one-variable maximum, the operational capacity (defined by codes with collective POVM decoding, not by regularization) equals the one-use Holevo value, and equiprobable phase-pair product ensembles attain it. This is the headline. Additivity with an arbitrary different partner channel, a secondary claim, is not formalized. Not rebuilt here.

Sources

Changelog1 change

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