Search Talks

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.

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 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).

Work on formulating general probabilistic theories in an operational context has tended to concentrate on the probabilistic aspects (convex cones and so on) while remaining relatively naive about how the operational structure is built up (combining operations to form composite systems, and so on). In particular, an unsophisticated notion of a background time is usually taken for granted. It pays to be more careful about these matters for two reasons. First, by getting the foundations of the operational structure correct it can be easier to prove theorems. And second, if we want to construct new theories (such as a theory of Quantum Gravity) we need to start out with a sufficiently general operational structure before we introduce probabilities. I will present an operational structure which is sufficient to provide a foundation for the probabilistic concepts necessary to formulate quantum theory. According to Bob Coecke, this operational structure corresponds to a symmetric monoidal category. I will then discuss a more general operational framework (which I call Object Oriented Operationalism) which provides a foundation for a more general probabilistic framework which may be sufficient to formulate a theory of Quantum Gravity. This more general operational structure does not admit an obvious category theoretic formulation.