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Shared Randomness and Normalization in 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 study a finite-message communication model defined by contracting one backward binary symmetric channel with one fixed, finite-dimensional forward quantum channel and then normalizing the resulting weights. The devices may share a finite classical seed that is independent of the message before normalization. The seed branches are mixed at the level of unnormalized weights, followed by a single normalization for each message. For M ≥ 2 messages, backward crossover probability 0 < p < 1/2, and forward channel Φ, we derive the exact worst-message success probability S_raw(Φ,p,M) = 2(1−p)/[M−(M−2)(1−2p)δ(Φ)], where δ(Φ) = max over input states ρ, σ of ½‖Φ(ρ)−Φ(σ)‖₁. The quantity δ is the established trace-distance contraction coefficient, not a new channel invariant. A cyclic construction with M seed values attains the bound. Crucially, normalization can make the posterior seed distribution message-dependent: the theorem does not describe an average of separately normalized branches or require success for every seed. We give an exact example separating these conventions. Matched depolarizing-qubit and classical binary symmetric forward channels consequently have equal optima in this resource model. The result is a restricted one-use optimization, not a physical backward-communication protocol or a general asymptotic capacity theorem.

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 screening, and manuscript drafting under Daniel A. Pyle’s direction. Independent human review of the model, proofs, bibliography, and originality remains 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 and SymPy; LaTeX/pdfLaTeX.

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Version history

  1. v1Submitted by Daniel Adam PyleInitial depositCurrentOct 31, 2026