Where quantum meets logic, . . . in a world of pictures!

Abstract

Yes, that's indeed where it happens. These pictures are not ordinary pictures but come with category-theoretic algebraic semantics, support automated reasoning and design of protocols, and match perfectly the developments in important areas of mathematics such as representation theory, proof theory, TQFT & GR, knot theory etc. More concretely, we report on the progress in a research program that aims to capture logical structures within quantum phenomena and quantum informatic tasks in purely diagrammatic terms. These picture calculi are faithful representations of certain kinds of monoidal categories, and structures therein. However, the goal of this program is partly to `release' these intuitive languages (or calculi) from their category-theoretic underpinning, and conceiving these pictures as mathematical entities in their own right. In this new language one is able to model and reason about things such a complementary observables, phase data, quantum circuits and algorithms, a variety of different quantum computational models, hidden-variable models, aspects of non-locality, and reason about all of these in terms of intuitive diagram transformations. Some recent benchmarks are the diagraamatic computation of quantum Fourier transform due to Duncan and myself, a purely diagrammatic proof of the no-cloning theorem due to Abramsky, and a categorical characterisation of GHZ-type non-locality due to Edwards, Spekkens and myself. For informal introductions we refer to: [1] Kindergarten quantum mechanics. http://arxiv.org/abs/quant-ph/0510032 [2] Introducing categories to the practicing physicist. http://arxiv.org/abs/0808.1032 For recent more advanced developments we suggest: [3] Selinger: Dagger compact closed categories and completely positive maps QPL\'05 http://www.mathstat.dal.ca/~selinger/papers.html#dagger [4] Coecke, Pavlovic, Vicary: A new description of orthogonal bases. http://arxiv.org/abs/0810.0812 [5] Coecke, Paquette, Perdrix: Bases in diagrammatic quantum protocols http://arxiv.org/abs/0808.1029 [6] Coecke, Duncan: Interacting quantum observables. ICALP\'08. http://www.springerlink.com/content/y443214116h76122/ [7] Coecke, Edwards: Toy quantum categories. QPL\'08. http://arxiv.org/abs/0808.1037