Model theory (a branch of mathematical logic) has yielded deep insights across mathematics through the study of type spaces and definability. In each of these settings, however, the underlying truth values remain classical -- two-valued in ordinary first-order logic, or [0,1]-valued in real-valued logic. This leaves a gap at the interface between model theory and the noncommutative, operator-algebraic semantics native to quantum theory, where the classical setting is deterministic and commutative while the quantum setting is not.

We introduce a new model-theoretic framework in which truth values are assigned not in {0,1} or [0,1], but in the projections of a von Neumann algebra. Logical formulas are interpreted as measurements, with truth assigned via quantum measures rather than by a classical valuation. Within this framework, classical model-theoretic tools now admit quantum-valued counterparts.

Recently, by combining Todorcevic's Trichotomy for Rosenthal compacta and Shelah's concept of NIP theories with Keisler's new real-valued logic, a team of researchers and students from UofT and UT San Antonio discovered a hierarchy of complexity classes in the (classical) floating-point computation setting. Because the model-theoretic tools behind that discovery now have quantum-valued counterparts within our framework, we expect to replicate this result in the quantum computing setting, as an application of the quantum-valued model theory we are proposing, yielding new, analogous complexity classes.

This is ongoing interdisciplinary work carried out by researchers and students at the University of Toronto, led by Frank Tall, and the University of Texas at San Antonio, led by Eduardo Dueñez and José Iovino. The extension to quantum computing is being developed by a separate group of students, including Olivia Aubone.