This course will explore applications of quantum information theory to the statistical modeling of classical dynamical systems. We will start by introducing operator-theoretic approaches to ergodic theory, which characterize dynamical systems through their induced action on linear spaces of observables. This action is through linear evolution operators—the Koopman operators, which act on observables by composition with the dynamical flow map, and the transfer operators, which are the dual operators acting on spaces of measures. For certain types of systems, these operators have very similar mathematical properties to the linear evolution operators for observables and states, respectively, of a quantum system. The second part of the course will explore these connections in some detail, using notions from the theory of von Neumann algebras to describe how the statistical evolution of a classical dynamical system can be embedded in an appropriately defined, infinite-dimensional quantum system. Discretization (i.e., finite-dimensional projection) of the infinite-dimensional system then leads to finite-dimensional quantum systems that approximate the original classical dynamics. We will study the properties of these finite-dimensional quantum systems, including their simulability by quantum computers.
A tentative weekly plan for the course can be found here.