A Mathematical Foundation For Locality
This work is motivated by two non-intuitive predictions of Quantum Mechanics: non-locality and contextuality. Non-locality is a phenomenon whereby interactions between spatially separated objects appear to be occurring faster than the speed of light. Contextuality is a phenomenon whereby the outcome of a measurement cannot be interpreted as the revelation of an intrinsic fixed property of the system being measured, but instead necessarily depends on the configuration of the measurement apparatus. Quantum Mechanics predicts non-local behavior in certain types of experiments collectively known as Bell tests. However, ruling out all possible alternative local theories is a subtle and demanding task. In this work, we lay out a mathematically-rigorous framework for analyzing Bell experiments. Using this framework, we derive the famous Clauser-Horne-Shimony-Holt (CHSH) inequality, an important constraint that is obeyed by all local theories and violated by Quantum Mechanics. We further demonstrate how to analyze the data of a CHSH experiment without assuming that successive experimental trials are independent and/or identically distributed. We also derive the Clauser-Horne (CH74) inequality, an inequality that is more well-suited for realistic Bell experiments using photons. We demonstrate a robust method for statistically analyzing the data of a CH74 experiment, and show how to calculate exact p-values for this analysis, improving on the previously-best-known (loose) upper bounds obtained from Hoeffding-style inequalities. The work concludes with an exploration of contextuality. The Kochen-Specker theorem -- a result demonstrating the contextual nature of Quantum Mechanics -- is applied to resolve a conjecture in Domain Theory regarding the spectral order on quantum states.