Quantum information geometry as a foundation for quantum theory beyond quantum mechanics

I will present a new approach to foundations of quantum theory that joins the insights of the maximum entropy approach to statistical mechanics (Jaynes, Ingarden), the information geometric approach to statistical inference (Chencov, Amari), and the operator algebraic approach to quantum theory (Araki, Haag). Nonlinear quantum kinematics is constructed using quantum information geometric structures over normal states on W*-algebras (quantum relative entropies and Banach Lie-Poisson structure), and replaces the foundational role played in von Neumann's quantum mechanics by probability theory, spectral paradigm, and Hilbert spaces. Nonlinear quantum dynamics is defined as constrained relative entropy maximisation (generalising Lüders’ rule) composed with nonlinear hamiltonian flow (generalising unitary evolution). It provides a prescription of noncommutative causal inference, and a nonlinear nonmarkovian alternative to the paradigm of completely positive instruments. Noncommutative Orlicz spaces play a special role in this new quantum kinematics and dynamics.