1 · The essentials
The question is how much ordinary information can be sent through a particular kind of noisy quantum link. The manuscript answers this for every link in the two-parameter family called qubit generalized amplitude damping, covering all damping strengths and thermal settings. It gives the answer by maximizing an explicit expression over just one variable. Only two pure quantum signal states are needed to reach the best information value for a single use. Independently chosen copies of those signals attain the corresponding optimum for any number of uses, with collective decoding used to achieve the communication capacity. The manuscript also establishes that pairing such a link with any other finite-dimensional quantum channel makes three key quantities—including the communication capacity—add rather than change unexpectedly. The main result has a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine sending messages through a noisy delivery service using just two kinds of cards. The result says that choosing these cards independently is enough for optimal communication, even though the receiver may need to interpret a whole batch together; the cards are only an analogy for quantum states.
Why it matters
The result turns the capacity question for this entire channel family into a one-variable optimization. It also connects the best single-use information value to the capacity achievable over many uses, rather than leaving a gap between them. The additivity statements apply even when the accompanying channel is any finite-dimensional quantum channel, not merely another member of the same family.
2 · Going further
The precise problem is to determine the regularized unassisted classical capacity of every qubit generalized amplitude-damping channel, for all damping strengths and thermal occupations. The main manuscript identifies this capacity with the one-shot Holevo capacity and expresses their common value as a one-variable maximum. The supplied abstract states that such a formula exists but does not provide its expression.
The optimization has a particularly small attaining ensemble: two pure states suffice for the one-use optimum. Independent products of this binary ensemble attain the optimum at every block length. Together with collective decoding, this supplies the stated route from the single-use optimization to the unassisted classical capacity. The abstract describes this structure of the result, but does not give the entropy inequalities or intermediate argument used to establish it.
There are also three additivity claims. If is a qubit generalized amplitude-damping channel and is any finite-dimensional completely positive trace-preserving channel, the stated identities are , , and . Here denotes one-shot Holevo capacity, minimum output entropy, and regularized unassisted classical capacity.
Only one manuscript is supplied, so there are no companion manuscripts to compare. It is identified as Lean-formalized, and the main result has a Lean formalization; the source does not give a claim-by-claim description of the formalization's coverage.
Useful notions
- Qubits and pure quantum states
- Quantum channels and noise
- Signal ensembles
- Entropy and Holevo capacity
- Tensor products and repeated channel uses
What was known before / what is new
Before
The source frames the problem using one-shot Holevo capacity, minimum output entropy, and regularized unassisted classical capacity. It does not describe earlier partial results, name a prior open problem, or provide historical context, so no specific earlier boundary of knowledge can be inferred from it.
New
The manuscript determines the unassisted classical capacity throughout the generalized amplitude-damping qubit family, with a one-variable optimization and a binary pure-state attaining ensemble. Its independent products attain the optimum at every block length. It also establishes additivity of three quantities when a channel in this family is paired with any finite-dimensional completely positive trace-preserving channel.
Limits and assumptions
The capacity determination concerns qubit generalized amplitude-damping channels and unassisted classical communication, although it includes all damping strengths and thermal occupations within that family. The partner in each additivity statement must be a finite-dimensional completely positive trace-preserving channel. Independent signal preparation should not be confused with independent decoding: the repository summary specifies collective decoding. The main result has a Lean formalization, but the supplied material does not specify which additional claims are individually covered. It also supplies neither the explicit optimization formula nor a detailed argument, and identifies no remaining open problem.
Mini glossary
- Unassisted classical capacity —
- The optimal rate of ordinary information transmission through a quantum channel without supplementary assistance.
- One-shot Holevo capacity —
- The optimal Holevo information of an ensemble of signals for a single use of a quantum channel.
- Binary pure-state ensemble —
- A signal collection consisting of two pure quantum states, each assigned a selection probability.
- Collective decoding —
- Decoding that processes outputs from multiple channel uses together.
- Minimum output entropy —
- The smallest entropy among the output states a channel can produce.
- Additivity —
- The property that a quantity for a tensor-product channel equals the sum of that quantity for its two component channels.
Keywords : Generalized amplitude damping · Classical capacity · Additivity
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Classical capacity and entropy inequalities for generalized amplitude damping
24/09/2026We 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.