Towards postquantum information relativity: a status report

In this talk I will: 1) review the results of my work on a geometric approach to foundations for a postquantum information theory; 2) discuss how it is related to other foundational approaches, including some resource theories of knowledge and quantum histories; 3) present some of my research on a category theoretic framework for a multi-agent information relativity. More details on part 1: this approach does not rely on probability theory, spectral theory, or Hilbert spaces. Normalisation of states, convexity, and tensor products are allowed but not assumed foundationally. Nonlinear generalisation of quantum kinematics and dynamics is constructed using geometric structures (quantum relative entropies and Banach Lie--Poisson structure) over the sets of quantum states on W*-algebras. In particular, unitary evolution is generalised to nonlinear hamiltonian flows, while Lueders' rules are generalised to constrained relative entropy maximisations. Combined together, they provide a framework for causal inference that is a generalisation and replacement for completely positive maps, with information dynamics determined directly by epistemic constraints, and no requirement for lack of correlation. Orthodox probability theory and quantum mechanics are special cases of this framework. I will also give the progress report on the reconstruction conjecture: given the category of sets of abstract "states" equipped with the suitably defined entropic distances and BLP structure, how one reconstructs the W*-algebraic case? The discussion of the consistent operational semantics for this approach will lead us to the parts 2 and 3.