Scale-Dependent Quantum Inference via Environmental Transformation Description This record contains the updated and finalized version of the paper titled Scale-Dependent Quantum Inference via Environmental Transformations. This version represents a unified and fully polished synthesis of the conceptual frameworks introduced in the author’s earlier works: Thought Experiment on Photon Pulse Reconstruction in Extreme Environments (Paper 1) Quantum Forensics: Inferring Photon States via Environmental Bow Waves (Paper 2) The updated paper formalizes the relationship between a photon’s initial state and an observable environmental signal through a transformation operator �, where the environment is treated as an active encoding medium rather than a passive source of noise. A central contribution of this version is the explicit integration of scale-dependent behavior via a complexity parameter �, which governs the degree to which environmental transformations preserve or obscure information about the underlying photon state. As environmental complexity increases, the transformation becomes increasingly non-invertible, leading to reduced mutual information and Fisher information between the photon state and the observable signal. The paper develops the interpretation of environmental responses as structured perturbations, described as “bow waves,” which represent distributed signatures of photon–medium interactions. These signatures form the basis of a quantum forensic approach, in which the photon state is inferred indirectly through its environmental imprint. A toy model is included to illustrate these principles in a simplified one-dimensional system consisting of a photon interacting with a chain of stochastic scatterers. The model demonstrates how recursive interactions and stochastic feedback lead to progressive information loss as system size increases. This updated version improves the clarity, internal consistency, and formal presentation of the framework. It refines terminology, strengthens the mathematical and conceptual structure, and aligns the exposition into a coherent, submission-ready document suitable for archival and dissemination. Version Notes This version supersedes the previous draft of the same work hosted under this record. The following updates have been incorporated: Full integration of conceptual material from related foundational works (Paper 1 and Paper 2) into a unified framework Refinement and standardization of terminology across the document, including consistent use of: “environmental transformations” “quantum forensics” “bow waves” “scale parameter �” Improved clarity and precision of language throughout the abstract, introduction, and analytical sections Strengthening of the information-theoretic formulation, including clearer articulation of mutual information and Fisher information as measures of recoverability Enhanced discussion of inference conditioning and ill-posedness in high-complexity regimes Expansion and clarification of the toy model to better illustrate recursive stochastic interactions and signal overlap behavior Addition and refinement of the limitations and scope section to explicitly contextualize the model’s purpose and applicability Overall polishing of structure, flow, and consistency to meet submission-ready standards This version should be considered the finalized and authoritative iteration of the work. Keywords Quantum inference Information theory Fisher information Mutual information Inverse problems Environmental transformations Quantum measurement Quantum forensics Signal processing Statistical inference Complex systems Photon interactions Stochastic processes Non-invertible systems Scale-dependent models Subjects Physics Quantum Physics Information Theory Statistical Physics Applied Mathematics Complex Systems License Creative Commons Attribution 4.0 International (CC BY 4.0)
Craig Kyrle Strachan Davidson (Tue,) studied this question.