Operational Signalling is Not Identifiable from Finite Measurement-Dependent Data

# Operational Signalling is Not Identifiable from Finite Measurement-Dependent Data **Author:** Bob Jefferson **ORCID:** https://orcid.org/0009-0003-1460-055X --- ## 📌 Overview This work develops a finite, explicit framework for analysing **operational signalling** in measurement-dependent latent-variable models. It establishes a sharp distinction between: - **Formal signalling** — dependence of observed distributions on remote settings - **Operational signalling** — the existence of a usable communication channel The central result is that these notions do not coincide: apparent signalling in observational data does not determine whether communication is possible. --- ## ✨ Main Results ### 1. Non-identifiability Operational signalling **cannot be inferred from finite observational data alone**. The same family of conditional distributions \( \{μ_x^y\} \) can be realized both:- with zero signalling capacity, and - with nonzero signalling capacity --- ### 2. Exact capacity law The maximal operational signalling strength satisfies: \[C_yᵐᵃˣ(Γ) = Γ D_y\] where:- \(D_y\) = observable distinguishability (total-variation diameter) - \(Γ\) = control parameter (message-to-setting influence) This factorises signalling into:> **observable distinguishability × latent control** --- ### 3. Binary structural constraint In a restricted symmetric binary subclass: \[Γ + Δ ≤ 1\] where:- \(Δ\) = latent measurement dependence This gives the operational bound: \[C_y ≤ (1 - Δ)\, D_y\] --- ### 4. Higher-dimensional limitation An explicit counterexample shows: - the binary tradeoff is **not universal** - for larger setting spaces, control and measurement dependence can both be maximal --- ## 🧠 Significance These results show that: - Observational data alone cannot determine whether a communication channel exists - Operational signalling depends on **latent causal structure**, not just observed correlations - Even so, structural constraints can emerge in restricted settings This has implications for: - Bell-type scenarios - measurement dependence - quantum communication and cryptography - the operational interpretation of signalling --- ## 📂 Files - **OperationalSignalling.pdf** — final paper - **OperationalSignalling.tex** — LaTeX source - **references.bib** — bibliography --- ## 📜 License This work is licensed under **Creative Commons Attribution 4.0 (CC BY 4.0)**.