Twisting Probabilities – An Introduction to Quantum Contextuality

In probability theory, one wants to assign probabilities to a set of events. This assignment must be compatible with the logical structure of the events – if A implies B, then the probability of B must be at least that of A. The classical approach to probability theory achieves this using measure theory. The events are modelled as a collection of subsets of some set, called the sample space. Each point (“outcome”) in the sample space assigns a truth value to every event at once. A probability measure then (at least morally) assigns a probability to each outcome, and thereby to each event. In this talk, I will demonstrate that this picture is incompatible with quantum mechanics. In the quantum formalism, some events are “incommensurable”, meaning they cannot be evaluated for truth simultaneously. Quantum mechanical predictions come in the form of “local” probability distributions defined only over collections of “commensurable” events – called contexts. In some cases, these local distributions cannot be glued together consistently to any underlying “global” distribution, even when they locally match. This result is known as quantum contextuality. In the talk, I will introduce the idea of contextuality at a friendly pace (with examples!) and show how it appears from the Hilbert space formalism of quantum mechanics. With this as motivation, I will argue that a framework for probability theory more general than measure theory is needed to allow contextual distributions, and time permitting, outline some existing approaches towards this end. This will not be an analysis heavy presentation, and no PDE:s will appear, but I hope it will give a fresh perspective on the relationships between measure theory, probability, functional analysis, and physics.

This initiative is part of the "PhD Lectures" activity of the project "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.