Interacting quantum observables
bob coecke, Oxford University Computing Laboratory
Date and time: Friday 7th November 2008 at 14:00
Location: UG40, School of Computer Science
Joint work with Ross Duncan, ICALP'08.
Within a high-level category-theoretic framework, as well as in terms of an intuitive purely diagrammatic calculus, we axiomatise an essential feature of quantum mechanics. This feature is the interaction of incompatible of quantum observables. A typical example is the interaction of position and momentum.
Axiomatic approaches to quantum mechanics have historically focussed on the failure of various properties: non-commutative algebras, non-distributive lattices, non-Kolmogorovian probabilities, etc. More recent attempts, originating in quantum computer science, concentrate mainly on accommodating no-cloning and no-deleting. Here we take a positive attitude: we identify the flows of information between incompatible observables.
We show that a pair mutually unbiassed quantum observables forms a bialgebra-like structure. Additional structure enables us to derive all observables and define a computationally universal system. The resulting equations suffice to perform intuitive computations with elementary quantum gates, translate between distinct quantum computational models, establish the equivalence of entangled quantum states, and simulate quantum algorithms such as the quantum Fourier transform.