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Classical capacity and entropy inequalities for generalized amplitude damping
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Abstract
We determine the unassisted classical capacity of every qubit generalized amplitude-damping channel, for all damping strengths and thermal occupations. It equals the one-shot Holevo capacity and is given by a one-variable maximum. A binary pure-state ensemble attains the one-use optimum, and its independent products attain the optimum at every block length. We also prove that one-shot Holevo capacity, minimum output entropy, and regularized unassisted classical capacity are additive under tensor product with every finite-dimensional completely positive trace-preserving partner channel.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Kenta Kitamura (GitHub: KitaKen1), qubit-strong-additivity (GitHub repository, Lean v4.34.1, draft of 11 Oct 2026). Claims, with a Lean proof, strong Holevo additivity for every qubit-to-qubit channel with any finite-dimensional partner, which would include the Holevo part of this paper's partner-additivity corollary; it lists Fang (2026) and OpenAI (2026) as earlier results for generalized amplitude damping. [AI-generated per source]
Formalization
- OpenAI math release, Lean scope note for family 276 (lean/docs/276.md, 6 Oct 2026). The note says the Lean development covers Holevo additivity over all tensor powers, the attained scalar maximum, operational capacity, and attainment by phase-pair product ensembles, while excluding partner-channel additivity and decoding by separate measurements. [family-level]
- OpenAI math release, comparator statement AmplitudeDamping.lean (lean/ComparatorChallenges). Lean statement of the main theorem: the n-use Holevo value is n times the one-use value and equals the scalar maximum, capacity equals the one-use Holevo value, and phase-pair ensembles attain it.
- QuAIR group, Classical-Capacity-Qubit-Amplitude-Damping (GitHub repository, Lean 4 + Lean-QIT). Lean formalization of the cited Tang–Zhu–Bai–Wang manuscript, covering the zero-temperature (ordinary amplitude-damping) case and the matrix entropy inequalities that this paper extends to the thermal case.
Related prior literature mentioned by others
- Z. Tang, C. Zhu, G. Bai, X. Wang, Classical Capacity and Entanglement Cost of the Amplitude Damping Channel (arXiv:2609.28592, 23 Sep 2026). Fang's v2 says its norm-compression proof extends this paper's log-determinant method; also cited in this paper as [TangEtAl2026].
Possibly relevant literature (found by the agent, not checked)
- V. Siddhu, A. Chatterjee, K. Jagannathan, P. Mandayam, S. Tayur, Queue-Channel Capacities with Generalized Amplitude Damping (arXiv:2107.13486, 2021). Derives the exact classical capacity of the symmetric generalized amplitude-damping channel (ν = 1/2), relevant to the paper's discussion of the unital point.
- Stefano Pirandola, Exact series formulas for the capacities of the amplitude damping channel (arXiv:2609.31609v1, 25 September 2026). Gives a convergent series for the unassisted classical capacity in the ordinary amplitude-damping boundary case of this paper's scalar optimization, with AI-assisted derivations disclosed by the author.
Digests and expositions
- Open Mathematics Courses, Classical capacity of generalized amplitude damping (undated). Five-lesson exposition of the manuscript's log-determinant and block-entropy argument through tensor-power and arbitrary-partner additivity, treating the classical coding theorem as a named hypothesis and stating that independent review is incomplete. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 276
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Classical-capacity-and-entropy-inequalities-for-generalized-amplitude-damping-September-24-2026,
author = {{OpenAI}},
title = {{Classical capacity and entropy inequalities for generalized amplitude damping}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Classical-capacity-and-entropy-inequalities-for-generalized-amplitude-damping-September-24-2026/paper.pdf}{OAI:Classical-capacity-and-entropy-inequalities-for-generalized-amplitude-damping-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/276.md. The Lean statements have not been checked by people against the paper's claims.
Family 276: Classical capacity of generalized amplitude damping
Determines the unassisted classical capacity of every qubit generalized amplitude-damping channel, including all damping and thermal parameters. An explicit one-variable optimization gives the capacity, attained by independent two-state signal ensembles with collective decoding. Holevo capacity, minimum output entropy and regularized classical capacity are additive when tensoring with any finite-dimensional quantum channel.
This is the only paper in this family.