Generalized possibilistic Theories: the tensor product problem
Résumé
Inspired by the operational quantum logic program, we have the contention that probabilities can be viewed as a derived concept, even in a reconstruction program of Quantum Mechanics. We propose an operational description of physical theories where probabilities are replaced by counterfactual statements belonging to a three-valued (i.e. possibilistic) semantic domain. The space of states and the space of effects are then built as posets put in duality through a Chu 3 space. The convexity requirements on the spaces of states and effects, addressed basically in Generalized Probabilistic Theories, are then replaced by semi-lattice structures on these spaces. The pure states are also easily constructed as completely meet-irreducible elements which generate the whole space of states. The channels (i.e. symmetries) of the theory are then naturally built as Chu morphisms. An axiomatic can then be summarized for what can be called "Generalized possibilistic Theory" based on this States/Effects Chu space's category. The problem of bipartite experiment is then addressed as the main skill of this paper. An axiomatic for the tensor product of the space of states is then given and a solution is explicitly constructed. The relations/differences between this tensor product and the tensor product of semi-lattices present in the mathematical literature are then analyzed. This new proposal for the tensor product of semi-lattices can be considered as an interesting byproduct of this work.
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