[This is a joint seminar with OASIS]
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).