Bob Coecke (University of Oxford)

`Bob.Coecke@comlab.ox.ac.uk`

Achieving both a foundational and high-level understanding of the quantum mechanical structure is a long-standing problem, ever since John von Neumann denounced his own quantum mechanical formalism back in 1935. This quest is today more relevant than ever in the light of the recent quantum informatic endeavour. It is fair to say that the current manipulations of matrices (i.e. arrays of complex numbers) are kin to the manipulations of 0's and 1's in the early days of computing.

We report on a recent research strand, initiated by Abramsky and myself in [1], and further developed for example in [2, 3, 4, 5, 6, 7]. We show that finite dimensional quantum mechanics itself supports a purely diagrammatic high-level quantum formalism. In fact, this diagrammatic formalism both formalizes and extends Dirac's bra-ket notation for quantum mechanics in a 2-dimensional fashion. Importantly, while most of the quantum structural research has been thus far `purely academic', the diagrammatic calculus proves to be extremely useful for the design and analysis of quantum information protocols, both qualitatively and quantitatively. For example, it turns several sophisticated quantum informatic protocols into trivial undergraduate exercises [4]. As compared to Birkhoff-von Neumann quantum logic, which has led to an order-theoretic paradigm for the study of the quantum mechanical structure, this new setting does come with traditional logical mechanisms such as deduction. In fact, it turns out to be some kind of hyper-logic as compared to the Birkhoff-von Neumann non-logic. The actual mechanism of deduction diagrammatically incarnates as `yanking a rope'. There are also strong connections of this work with other fields of mathematical physics such as topological quantum field theory and knot theory.

The main recent development in this research program is the ability to capture quantum measurements and classical data manipulations within the language which was initially designed to capture quantum entanglement [5, 6, 7]. We have:

quantum | linear logic |

classical | classical logic |

We are able to distinguish between classical non-determinism, stochastic processes, reversible classical processes etc. At the core of all this lies an analysis of the abilities to clone and delete data in the classical world `from the perspective in the quantum world'. In this view, the classical world looks surprisingly complicated as compared to the very simple quantum world.

**References**

[1] S. Abramsky and B. Coecke (2004). A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, pp.415-425. IEEE Computer Science Press. quant-ph/0402130.

[2] B. Coecke (2005). De-linearizing Linearity: Projective Quantum Axiomatics from Strong Compact Closure. quant-ph/0506134

[3] P. Selinger (2005). Dagger compact closed categories and completely positive maps. http://www.mathstat.dal.ca/~selinger/papers.html#dagger

[4] B. Coecke (2005). Kindergarten quantum mechanics - lecture notes. quant-ph/0510032

[5] B. Coecke and D. Pavlovic (2006). Quantum measurements without sums. quant-ph/0608035

[6] B. Coecke and E. Paquette (2006). POVMs and Naimark's theorem without sums. quant-ph/0608072

[7] B. Coecke, E. Paquette and D. Pavlovic (2007). Classical structures from tensorial quantum structures. Draft paper.