Kindergarten quantum mechanics: beyond the Hilbert space formalism

We discuss a graphical language (and the corresponding formal algebra) which supports high-level quantum reasoning. Physicists should welcome this language since it both refines and formalises the highly successful Dirac calculus in a very intuitive two- dimensional fashion. Computer Scientists should welcome it since its algebra is in fact an extremely powerful logical system which enables automated design and verification. The main recent development in this research program is the ability to capture quantum measurements, classical data manipulations, phase data and quantum informatic quantities and concepts within the language which was initially designed to reason about quantum entanglement. We are for example 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. This is a lesson computer scientists already knew about for a while: classical logic becomes much easier to manipulate if you decompose it as: classical logic = linear logic + (copying, deleting).

We can now do computations which some sceptics never imagined possible within our purely graphical calculus e.g.:

• derive dense coding and teleportation schemes including classical control 
• prove Naimark's extension theorem for POVMs 
• establish resource inequalities involving coherent communication 
• prove universality of measurement based computational schemes 
• compute the quantum Fourier Transform

Ross Duncan will discuss the latter two in his talk which follows mine. Several people contributed to this research program including Samson Abramsky, Ross Duncan, Dusko Pavlovic, Eric Paquette and Peter Selinger. For a somewhat informal introduction to this research program please consult quant-ph/0510032 and quant-ph/0506132.