The topology of Dirac's notation, and beyond.
by
Bob Coecke
Oxford University

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 in [2,3,4,5,6,7,8]. While traditionally only infinite dimensional Hilbert spaces have been considered to be of interest to topologists, we show that finite dimensional quantum mechanics itself supports a purely topological high-level quantum formalism. In fact, this topological 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 topological calculus proves to be extremely useful for the design and analysis of quantum information protocols, both qualitatively and quantitatively [1]. For example, it turns several sophisticated quantum informatic protocols into trivial undergraduate exercises [4]. Also theorems such as Naimark's theorem admit extremely elegant purely topological proofs [6]. 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 super-logic as compared to the Birkhoff-von Neumann non-logic. The actual mechanism of deduction topologically 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 [7]. We point to open problems where typical methods from domain theory such as recursive types and other methods of asymmetric topology could be of use.

[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] S. Abramsky and B. Coecke (2005) Abstract physical traces. Theory and Applications of Categories 14: 111-124.

[3] P. Selinger (2006) Dagger compact closed categories and completely positive maps. ENTCS (To appear). http://www.mathstat.dal.ca/~selinger/papers.html#dagger

[4] B. Coecke (2005) Kindergarten quantum mechanics - lecture notes. In: Quantum Theory: Reconsiderations of the Foundations III, pp.81--98. AIP Press. quant-ph/0510032

[5] B. Coecke (2005) Introducing categories to the practising physicist. In: What is category theory? Polimetrica Publishing. Advanced Studies in Mathematics and Logic 30, pp.45-74. Polimetrica Publishing.

[6] S. Abramsky (2005) Abstract scalars, loops, and free traced and strongly compact closed categories. LNCS 3629, 1-31, 2005.

[7] B. Coecke and D. Pavlovic (2005) Quantum measurements without sums. In: Mathematics of Quantum Computing and Technology. Taylor and Francis. (to appear)

[8] B. Coecke and E. O. Paquette (2006) Generalized measurements and Naimarks theorem without sums. ENTCS (To appear).

Date received: June 16, 2006