12 April 2005 4:00 - 5:30 Bob Coecke Abstract Quantum Mechanics III: De-linearizing linearity Abstract: We unravel the linear structure of Hilbert spaces into several constituents which turn out to play manifestly different roles in quantum mechanics. Some prove to be very crucial for particular features of the theory while others obstruct the passage to a formalism which is not saturated with insignificant global phases, hence these constituents need to be abandoned. This also means that our endavour entails a passage from a vector space like formalism to a more projective one as intended in the (in)famous Birkhoff-von Neumann paper [1]. Our study and constructions take the categorical formalism for quantum mechanics introduced by Samson Abramsky and myself in [2,3] as a starting point. The bulk of the required linear structure arises from the strong compact closed structure of the tensor [3] --- which compares to Kelly's notion of compact closure [4] as an inner-product space compares to a vector space --- which provides a variety of notions such as entangled states, scalars and positivity thereoff, unitarity and self-adjointness, trace, positive morphisms, Hilbert-Schmidt norm and Hilbert-Schmidt inner-product etc. Highly significantly is the fact that the passage from a formalism of the vector space kind to a more projective formalism takes place at the level of this tensor structure. Additive types for objects including a zero object provide pseudo projections from which measurements can be build, and the correctness proofs of the protocols discussed in [2] carry over to this much weaker setting. A full probabilistic calculus is obtained when the canonical trace [5] is moreover pseudo linear and satisfies the diagonal axiom. This work will soon be available [6]. [1] Birkhoff, G. and von Neumann, J. (1936) The Logic of Quantum Mechanics. Annals of Mathematics 37, 823--843. [2] Abramsky, S. and Coecke, B. (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS`04), IEEE Computer Science Press. (extended version including proofs at arXiv:quant-ph/0402130) [3] Abramsky, S. and Coecke, B. (2005) Abstract physical traces. Theory and Applications of Categories 14, To appear very soon. [4] Kelly, G. M. and Laplaza, M. L. (1980) Coherence for compact closed categories. Journal of Pure and Applied Algebra 19, 193-213. [5] Joyal, A., Street, R. and Verity, D. (1996) Traced monoidal categories. Proceedings of the Cambridge Philosophical Society 119, 447-468. [6] Coecke, B. (2005) De-linearizing linearity. Research Report, Oxford University Computing Laboratory, To be available very soon.