Result 276, Mathematical physics

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.

Lean formalization Classification or exact value

The bigger picture

Why it matters

Quantum noise can limit how much ordinary information a quantum system carries. This manuscript claims an exact capacity formula for a family of noisy two-level systems, together with limits on what joint encoding can gain.

What changes?

The manuscript covers every qubit generalized amplitude-damping channel, a noise model for a two-level quantum system with damping and thermal parameters, across all values of those parameters. It reports that the unassisted classical capacity equals the one-shot Holevo capacity, an information bound for one channel use, computable by maximizing over one variable. Two pure input states attain that optimum. Independent products of this ensemble remain optimal at every block length, with capacity achieved through collective decoding of outputs together.

What does that help mathematicians do?

The claimed additivity extends beyond repeated uses of the same channel: pairing it with any finite-dimensional quantum channel makes both the Holevo capacity and the regularized unassisted classical capacity equal the sums of the separate values. Minimum output entropy, the least uncertainty remaining in an output state, likewise adds. Researchers could therefore rule out capacity gains from joint encoding across such a pair and deduce its minimum output entropy from the components.

Are there practical applications?

The immediate value is foundational for quantum communication theory: the reported formula reduces an optimization over arbitrarily long transmissions to a single-variable calculation for this noise family. It could provide capacity limits for systems described by generalized amplitude damping. It does not itself provide a practical collective decoder or establish performance for hardware with other noise.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Classical capacity and entropy inequalities for generalized amplitude damping

September 24, 2026 20 pages Main result formalized in Lean

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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Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/276.md.

Classical capacity of generalized amplitude damping

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized result determines the unassisted classical capacity of every generalized amplitude-damping channel with parameters in [0,1]2[0,1]^2. Its unrestricted Holevo information is additive across every number of repeated uses, equals an attained scalar maximum, and agrees with the operational capacity per use. Every maximizing equiprobable phase-pair product ensemble attains the corresponding block value. Additivity with an arbitrary different partner channel and decoding by separate measurements are not included.

Comparator links

Result Comparator statement
Additivity and capacity for generalized amplitude damping AmplitudeDamping.lean

Posts about this result

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.