# Thermal qubit channel capacity calculator

Second edition, 9 October 2026 Australia/Brisbane. Initial decision: **Prototype numerical benchmark**. A working floating-point calculator now exists. Buyer demand, independent theorem verification and certified numerical error remain open.

## Research finding

Family 276 states that every memoryless qubit generalized amplitude-damping channel has unassisted classical capacity equal to an attained scalar Holevo maximum. It gives equiprobable pure-state phase pairs and product ensembles. Its selected formal scope covers repeated-channel additivity and capacity, excluding arbitrary-partner additivity and separate-output measurement decoding. [Manuscript](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Classical-capacity-and-entropy-inequalities-for-generalized-amplitude-damping-September-24-2026/paper.pdf), [selected formal scope](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/276.md).

## Problem and buyer

A quantum-communication R&D team needs a reproducible model-capacity reference after estimating damping and thermal population. Generic ensemble searches can be replaced by this explicit one-dimensional objective if the model and source result are accepted. A device team still needs to assess whether its observed channel fits the model.

## What the finding could enable

A channel-characterization plugin can provide the model objective, a candidate maximizing signal population and sensitivity across fitted parameter ranges. A custom lab application could compare measured communication rates to the claimed asymptotic reference while preserving the collective-decoding assumption. The calculator is small enough to be an open benchmark within a larger characterization system.

## Technical and commercial limits

The quantity is classical information capacity in bits per use, not quantum or secret-key capacity. The theorem assumes a memoryless channel and allows collective decoding. It does not provide a practical finite-block decoder. The current optimizer uses ordinary Python floating point, a complete finite grid and local refinement. It reports a conservative analytic grid-discretization allowance evaluated numerically, without an outward-rounded floating-point error bound. The result is not a formally certified capacity interval.

## Minimal architecture

Parameter validation -> stable binary entropy -> scalar grid search -> local candidate refinement -> grid-discretization allowance -> explicit evidence labels -> channel-fit and sensitivity report. The thermal parameter `nu` is the stationary **excited-state** population; the signal parameter `p` is the excited-state population of `sqrt(1-p)|0> ± sqrt(p)|1>`.

For the average output population `t=(1-gamma)p+gamma nu`, the output determinant is `v=gamma nu(1-nu)+gamma(1-gamma)(p-nu)^2`. The objective is the entropy of the average output minus the entropy of either phase-pair output, in bits. The exact manuscript formula and channel definition were inspected before implementing it.

## Existing alternatives and differentiation

Qiskit Aer already supplies a generalized amplitude-damping noise-channel constructor with Kraus matrices. That is a concrete integration target. Its documented channel construction and the new capacity objective serve different tasks; the limited documentation inspection does not prove the whole ecosystem lacks capacity calculations. The proposed value is a source-linked capacity benchmark and model-fit review. [Qiskit Aer amplitude-damping API](https://qiskit.github.io/qiskit-aer/stubs/qiskit_aer.noise.amplitude_damping_error.html).

## Monetization hypothesis

Prefer an open benchmark plus funded characterization integration, provisionally AUD 3,000–8,000 for a narrowly scoped project. At a hypothetical AUD 5,000 fee, 20 specialist hours at AUD 180/hour consume AUD 3,600 before support, sales and overhead. A standalone subscription is weak unless a buyer needs repeated channel fitting and reporting. No customer contracts or willingness-to-pay evidence exist.

## Validation experiment

The prototype passed 33 analytic or direct-matrix checks: identity channels, complete replacement channels, unital channels, thermal-population symmetry, agreement with a directly constructed two-by-two output matrix and coarse-grid allowances compared with finer searches. Across three parameter pairs, none of 1,500 sampled arbitrary pure-state ensembles exceeded the scalar benchmark. This sampling is a diagnostic, not a proof of optimality.

For `gamma=0.55` and `nu=0.3`, the best observed objective is approximately `0.3632124112` bits per use at `p≈0.477789669`. The 4,096-interval grid allowance is about `0.05745` bits; this conservative allowance shows that many reported decimal places are not a certified error guarantee. A tighter validated arithmetic backend would be needed for a rigorous narrow interval.

## Conditions to reject or defer

Reject a broad channel-design product if measured noise is outside the model, customers need finite-block decoding, or an existing tool already supplies the desired report. Defer certified numerical intervals until rounding is controlled and defer any formally verified label until a trusted independent check runs.

## Next concrete action

Add parameter-fit uncertainty, a documented solver connector and a validated arithmetic backend if a real user needs certification. Measure whether the benchmark changes a characterization decision before commercial packaging.

## Runnable prototype and saved evidence

- [Calculator](../../tools/gad_capacity.py).
- [Validation driver](../../tools/validate_gad_v1.py).
- [Saved validation](../../snapshots/2026-10-08-baseline/gad-validation-2026-10-09-v1.json).
- [Example output](../../snapshots/2026-10-08-baseline/gad-example-2026-10-09-v1.json).

From the workspace root:

```sh
python3 research/tools/gad_capacity.py --gamma 0.55 --nu 0.3
```

When saving outputs, pass a fresh path with `--output`. Existing files are preserved through exclusive creation.