Categories, Quanta, Concepts (CQC) - 2009

We give a mathematical framework to describe the evolution of quantum systems subject to finitely many interactions with classical apparatuus and with each other. The systems in question may be composed of distinct, spatially separated subsystems which evolve independently, but may also interact. The evolution is coded in a mathematical structure in such a way that the properties of causality, covariance and entanglement are faithfully represented. The key to this scheme is to use a special family of spacelike slices -- we call them locative -- that are not so large as to permit acausal influences but large enough to capture nonlocal correlations. I will briefly describe how the dynamics can be described as a functor to a suitable category of Hilbert spaces and will also give some connections with logic.

We sketch some ideas about how higher-dimensional categories could be used to extend conventional quantum mechanics. The physical motivation comes from quantum field theory, for which higher-dimensional category theory is very relevant. We discuss how this new approach would affect familiar aspects of quantum theory, such as observables and the Copenhagen interpretation. Few solid answers will be given, but hopefully some discussion will be generated!

Tensor product is described in a family of categories that includes Set and Hilbert spaces. Such categories admit a "scalar" object which enables a definition of bi-arrows with two domains, generalizing functions of two variables. The tensor product is characterized by the expected universal property relating bi-arrows to arrows.

The notion of a conditional probability is critical for Bayesian reasoning. Bayes’ theorem, the engine of inference, concerns the inversion of conditional probabilities. Also critical are the concepts of conditional independence and sufficient statistics. The conditional density operator introduced by Leifer is a natural generalization of conditional probability to quantum theory. This talk will pursue this generalization to define quantum analogues of Bayes' theorem, conditional independence and sufficient statistics. These can be used to provide simple proofs of certain well-known results in quantum information theory, such as the isomorphism between POVMs and convex decompositions of a mixed state and the remote collapse postulate, and to prove some novel results on how to pool quantum states. This is joint work with Matt Leifer. I will also briefly discuss the possibility of a diagrammatic calculus for classical and quantum Bayesian inference (joint work with Bob Coecke).