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Models of quantum phenomena

Bob Coecke ( Computing Laboratory, University of Oxford )

A formulation of quantum mechanics in terms of symmetric monoidal categories provides a logical foundation as well as a purely diagrammatic calculus for it. This approach was initiated in 2004 in a joint paper with Samson Abramsky (Ox). An important role is played by certain Frobenius comonoids, abstract bases in short, which provide an abstract account both on classical data and on quantum superposition. Dusko Pavlovic (Ox), Jamie Vicary (Ox) and I showed that these abstract bases are indeed in 1-1 correspondence with bases in the category of Hilbert spaces, linear maps, and the tensor product. There is a close relation between these abstract bases and linear logic. Joint work with Ross Duncan (Ox) shows how incompatible abstract basis interact; the resulting structures provide a both logical and diagrammatic account which is sufficiently expressive to describe any state and operation of "standard" quantum theory, and solve standard problems in a non-standard manner, either by diagrammatic rewrite or by automation.

But are there interesting non-standard models too, and what do these teach us? In this talk we will survey the above discussed approach, present some non-standard models, and discuss in how they provide new insights in quantum non-locality, which arguably caused the most striking paradigm shift of any discovery in physics during the previous century. The latter is joint work with Bill Edwards (Ox) and Rob Spekkens (Perimeter Institute).

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