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The Logic of Entanglement. An invitation. (Version 0.9999)

Bob Coecke


In SHORT: We expose the information flow capabilities of quantum entanglement. In LONG: This paper contains several components: (a) We prove a general characterization theorem on information flow through bipartite entanglement. This theorem will enable us to provide a unified view on protocols such as quantum teleportation, quantum logic gate teleportation and entanglement swapping. (b) We accomplish the extension of the above to multipartite entanglement which exposes the necessity of logical tools such as typing. Also the need for linear logic connectives and polarities arises naturally. (c) We expose a methodology emerging from our information flow based reasoning about entanglement which yields a two-way compilation scheme enabling design of computational and communicational protocols. This tool allows evident reconstruction of the above mentioned protocols of quantum information processing and also the design of new ones in terms of a classical travelling token-interpretation. We use this methodology to realize a passage from sequential to parallel composition for quantum logic gates. This mechanism also yields a fault-tolerant methodology to prepare multipartite entangled states. (d) At a more advanced level this methodology allows to accommodate classical functional programming features such as Currying, lambda-calculi, Abramsky style geometry of interaction in the sense of and other high level specification logics. (e) Finally, the information flow capabilities of entanglement exposed in this paper yield a canonical family of entanglement measures for multipartite systems. They also provide an interpretation in terms of information flow capabilities for non-local untary operations.

Oxford University Computing Laboratory