Contextual Measurement Model and Quantum Theory

TL;DR

The paper presents the Contextual Measurement Model (CMM), a unified probabilistic framework that treats probability updates as context-dependent changes in preparation and measurement, thereby integrating classical, quantum, and semi-classical physics. It formalizes a contextual probability space and contextual instruments, deriving a generalized total-probability formula with an interference term $\delta_C(B=y|A)$ that explains quantum-like interference, order effects, and replicability within a single scheme. The work analyzes CMM across several realizations: Kolmogorov theory (classical), von Neumann observables, quantum instruments (POVMs and quantum channels), and ordered space representations, highlighting when Bell-type inequalities can be violated and how entanglement can be framed contextually without assuming tensor-product structure. It further discusses interpretations of contextual probability (frequency, ensemble, Kolmogorov, QBism-related perspectives) and extends the framework to quantum-like modeling in cognition and decision making. Overall, CMM reframes quantum probabilistic phenomena as context-driven probability updates, offering a versatile toolkit for foundational clarification and cross-disciplinary applications.

Abstract

We develop the contextual measurement model (CMM) which is used for clarification of the quantum foundations. This model matches with Bohr's views on the role of experimental contexts. CMM is based on contextual probability theory which is connected with generalized probability theory. CMM covers measurements in classical, quantum, and semi-classical physics. The CMM formalism is illustrated by a few examples. We consider CMM framing of classical probability, the von Neumann measurement theory, the quantum instrument theory. CMM can also be applied outside of physics, in cognition, decision making, and psychology, so called quantum-like modeling.