# 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 \ (\ₓʸ\ \) can be realized both: - with zero signalling capacity, and - with nonzero signalling capacity --- ### 2. Exact capacity law The maximal operational signalling strength satisfies: ᵧ^ () = Dᵧ\ where: - \ (Dᵧ\) = observable distinguishability (total-variation diameter) - \ (\) = control parameter (message-to-setting influence) This factorises signalling into: > **observable distinguishability × latent control** --- ### 3. Binary structural constraint In a restricted symmetric binary subclass: \ + 1\ where: - \ (\) = latent measurement dependence This gives the operational bound: ᵧ (1 -) \, Dᵧ\ --- ### 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) **.
Bob Jefferson (Tue,) studied this question.