Bob Coecke

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• [A partial order on classical and quantum states] A both quantitative and qualitative theory of information is developed which combines Shannon theory, a purely quantitative theory of information, and Scott's Domain Theory, a purely qualitative theory of information. We did this both for the cases of classical and quantum information. Important applications include the analysis of noise, semantics for probabilistic and quantum programs, and information security.
• [Partiality in Physics] We revisit the standard axioms of domain theory with emphasis on their relation to the concept of partiality, explain how this idea arises naturally in probability theory and quantum mechanics, and then search for a mathematical setting capable of providing a satisfactory unification of the two.
• [Entropic Geometry from Logic] We produce a probabilistic space from logic, both classical and quantum, which is in addition partially ordered in such a way that entropy is monotone. In particular do we establish the following equation: Quantitative Probability = Logic + Partiality of Knowledge + Entropy. This holds both for classical and quantum probability.