18/05/2004
Bob Coecke (Oxford University)
Quantum, concretely, abstractly

“Quantum” stands for for the concepts (both operational and formal) which had to be added to classical physics in order to understand otherwise unexplainable observed phenomena such as the structure of the spectral lines in atomic spectra. While the basic part of classical mechanics deals with the (essentially) reversible dynamics, quantum required adding the notions of “measurement” and (possibly non-local) “correlations” to the discussion. The corresponding mathematical formalism was considered to have reached maturity in [von Neumann 1932], but there are some manifest problems with that formalism:

1. While measurements are applied to physical systems, application of their formal counterpart (i.e. a self-adjoint linear operator) to the vector representing that state of the system in no way reflects how the state changes during the act of measurement. Analogously, the composite of two self-adjoint operators has no physical significance while in practice measurements can be effectuated sequentially. More generally, the formal types in von Neumann's formalism do not reflect the nature of the corresponding underlying concept at all!
2. Part of the problem regarding the measurements discussed above is that in the von Neumann formalism there is no place for storage, manipulation and exchange of the classical data obtained from measurements. Protocols such as quantum teleportation involving these cannot be given a full formal description.
3. The behavioral properties of quantum entanglement which for example enable continuous data exchange using only finitary communication are hidden in the formalism.

In [Abramsky and Coecke 2004] we addressed all these problems, and in addition provided a purely categorical axiomatization of quantum mechanics. The concepts of the abstract quantum mechanics are formulated relative to a strongly compact closed category with biproducts (of which the category FdHilb of finite dimensional Hilbert spaces and linear maps is an example). Preparations, measurements, either destructive or not, classical data exchange are all morphisms in that category, and their types fully reflect their kinds. Correctness properties of standard quantum protocols can be abstractly proven, and in this seemingly purely qualitative setting even the quantitative Born rule arises. Taking FdHilb as this category provides a concrete model of quantum mechanics. In particular, the properties exposed in [Coecke 2003] are perfectly captured by the compact closuredness of FdHilb.

S. Abramsky and B. Coecke. A categorical semantics of quantum protocols. In the proceedings of LiCS'04 (2004). An extended version is available at arXiv:quant-ph/0402130

B. Coecke. The logic of entanglement. An invitation. PRG-RR-03-12 (2003). An 8 page short version is at arXiv:quant-ph/0402014

J. von Neumann. Mathematische Grundlagen der Quantenmechanik. Springer-Verlag (1932). English translation in Mathematical Foundations of Quantum Mechanics. Princeton University Press (1955).