Talks

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.

Astronomers believe our Universe began in a Big Bang, and is expanding around us. Brian Schmidt will describe the life of the Universe that we live in, and how astronomers have used observations to trace our Universe's history back more than 13 Billion years. With this data a puzzling picture has been pieced together where 96% of the Cosmos is made up of two mysterious substances, Dark Matter and Dark Energy. These two mysterious forms of matter are in a battle for domination of the Universe, and Schmidt will describe experiments that are monitoring the struggle between Dark Energy and Dark Matter, trying better understand these elusive pieces of our Universe, and predict the ultimate fate of the Cosmos.

This talk will present an overview of work done in the past decade on quantum state and process tomography, describing the basic notions at an introductory level, and arguing for a pragmatic approach for data reconstruction. The latest results include recent numerical comparison of different reconstruction techniques, aimed at answering the question: "is 'the best' the enemy of 'good enough'?"

The transformation of a narrow beam into a hollow cone when incident along the optic axis of a biaxial crystal, predicted by Hamilton in 1832, created a sensation when observed by Lloyd soon afterwards. It was the first application of his concept of phase space, and the prototype of the conical intersections and fermionic sign changes that now pervade physics and chemistry. But the fine structure of the bright cone contains many subtle features, slowly revealed by experiment, whose definitive explanation, involving new mathematical asymptotics, has been achieved only recently, along with definitive experimental test of the theory. Radically different phenomena arise when chirality and absorption are incorporated in addition to biaxiality.

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).