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Finite Alphabet Communication with Noisy Forward Memory in a Normalized Loop

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

Contributed by Daniel Adam Pyle ↗

Version 1 / Oct 31, 2026 / CC BY 4.0

Abstract

We study one-use communication of a finite classical message through a stipulated normalized loop containing a backward binary symmetric channel and one noisy forward memory carrier. The carrier is either a classical bit or a qubit; no shared random seed, retained side register, or entanglement is supplied. For every finite message count, we characterize the qubit optimum by a constructive scalar feasibility test. For rational noise parameters and a rational target success, the test uses O(M²) constant-size rational arithmetic checks. The proof combines an exact decoder reduction, minimum-trace allocation for constant encoders, and a concave geometric cap with a matching polygon construction. For three messages, the qubit and classical-bit optima coincide throughout the allowed noise range; an explicit Bernstein certificate proves the remaining polynomial inequality. For four messages, an exact rational qubit protocol exceeds a bound valid for every classical-bit protocol. These are conditional mathematical results for the declared resource model, not a construction of a physical backward channel. Independent human review and originality assessment remain pending.

Provenance statement

This manuscript was prepared through an AI-assisted research workflow. Lyra, an OpenAI AI assistant, carried out the technical derivations, computational verification, literature checking, and manuscript drafting under Daniel A. Pyle’s direction. Independent human proof review and a fuller originality assessment remain pending. The results concern a stipulated mathematical model and do not establish physical backward communication.

Tools used

OpenAI
Lyra (OpenAI AI assistant); Python 3 with NumPy, SymPy and SciPy; LaTeX/pdfLaTeX.

References

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Version history

  1. v1Submitted by Daniel Adam PyleInitial depositCurrentOct 30, 2026