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Generalized possibilistic theories: the multipartite experiments problem

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Abstract

In a recent paper, the author introduced an operational description of physical theories where probabilities are replaced by counterfactual statements belonging to a three-valued (i.e., possibilistic) semantic domain. The complete axiomatic of these generalized possibilistic theories is generalized and clarified in the present paper. The problem of bipartite experiments is then addressed as the main skill of this paper. An axiomatic for the tensor product of our spaces of states is given and different solutions are explicitly constructed. This description of tensor products of Inf semi-lattices is partly independent from the usual mathematical description of this problem. The nature of the tensor product of ortho-complemented Inf semi-lattices is then also explored. This subject is indeed fundamental for the development of a reconstruction program for quantum theory within our framework. Our analysis constitutes a first step toward this achievement.

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Notes

  1. The objects of this category are the natural space of states in quantum mechanics, i.e., the Hilbert spaces of dimension greater than two, and the morphisms are the orbits on semi-unitary maps (i.e., unitary or anti-unitary) under the U(1) group action, which are the relevant symmetries of Hilbert spaces from the point of view of quantum mechanics.

  2. In the rest of this paper we refer to this construction, based on a three-valued Chu space, as a ’possibilistic’ approach to distinguish it from the ’probabilistic’ one.

  3. The information corresponding to macroscopic events/operations describing the procedure depend on an observer O. If this dependence has to be made explicit, we will adopt the notation \({ {\mathfrak {P}}}^{{}^{(O)}}\) to denote the set of preparation processes defined by the observer O. This mention of the observer will be also attached to the different quotients associated to the space of preparations.

  4. If the dependence with respect to the observer O has to be made explicit, we will adopt the notation \({ {\mathfrak {T}}}^{{}^{(O)}}\) to denote the set of tests defined by the observer O. This mention of the observer will be also attached to the different quotients associated to the space of yes/no tests.

  5. We note that complete meet-irreducibility implies meet-irreducibility. In other words,

    $$\begin{aligned} \sigma \in { {\mathfrak {S}}}_{{}_{\textrm{pure}}}\Rightarrow & {} (\, \forall \sigma _1, \sigma _2\in { {\mathfrak {S}}}, \;\;(\, \sigma = \sigma _1 \sqcap _{{}_{ { {\mathfrak {S}}}}} \sigma _2\,) \;\; \Rightarrow \;\; (\, \sigma = \sigma _1 \;\;\text { or}\;\; \sigma = \sigma _2 \,) \,). \end{aligned}$$
    (50)

  6. Throughout this short axiomatic introduction, we adopt the inadequate notation \(\boxtimes \) for the tensor product in order to allow for different candidates for this tensor product. These different candidates will be denoted \(\otimes \), \(\widetilde{\otimes }\),...

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  1. Université de la Côte d’Azur, CNRS, InPhyNi, Nice, France

    Eric Buffenoir

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Buffenoir, E. Generalized possibilistic theories: the multipartite experiments problem. Quantum Stud.: Math. Found. 10, 443–482 (2023). https://doi.org/10.1007/s40509-023-00306-3

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