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The Shared Randomness Cost of Optimal One Use Loop Communication

Primarily AI-generated textHuman understanding: no partsmath.IT — Information Theoryquant-ph — Quantum Physics

Contributed by Daniel Adam Pyle ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

We determine the minimum support and Shannon entropy of a shared classical seed needed to attain optimal worst-message success in a specified one-use normalized-loop model. The loop contains a backward binary symmetric channel with crossover 0 < p < 1/2 and a fixed finite-dimensional forward quantum channel. Seed branches are mixed as unnormalized weights before one normalization for each message. For qubit output and interior forward trace-distance diameter, equality at the known unrestricted optimum forces every supported seed to activate exactly two message labels. The seed-cost problem becomes a fractional perfect matching problem: even message counts are optimally covered by disjoint pairs, while odd counts require a triangle and disjoint pairs. We give separate exact formulas at zero and unit diameter, prove a sufficient higher-dimensional extension of the interior result, and exhibit an erasure-channel counterexample to diameter-only seed costs in general dimension. Explicit near-optimal constructions show how the exact-attainment cost discontinuities can be avoided at finite tolerance. Compactness also gives a strict performance gap below the minimum seed support. A complete exact-certificate calculation determines the two-seed classical optimum for three messages at backward and forward crossover 1/10. These are resource-specific mathematical results, not a physical backward-communication protocol or an asymptotic capacity theorem.

Provenance statement

Lyra, an OpenAI AI assistant, performed technical derivation, proof checking, code development and verification, bounded literature screening, and manuscript drafting in a workflow directed by Daniel A. Pyle. Separate AI checking passes are not independent human peer review. Independent human review and originality assessment remain pending.

Tools used

OpenAI
Lyra (OpenAI AI assistant); Python 3, including exact rational arithmetic, NumPy and SciPy; LaTeX/pdfLaTeX.

References

  1. D. A. Pyle, ``Shared randomness and normalization in one-use loop communication,'' unpublished research manuscript, 30 October 2026. Earlier computational material: https://github.com/dapyle11/normalized-loop-memory, archive shared_seed_loop_reproducibility (1).zip .link
  2. D. A. Pyle, ``Finite alphabet communication with noisy forward memory in a normalized loop,'' unpublished research manuscript, 30 October 2026. Related materials: https://github.com/dapyle11/normalized-loop-memory.link
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Version history

  1. v1Submitted by Daniel Adam PyleInitial depositCurrentOct 31, 2026