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Further on Non-Cartesian Systems

Elemér Elad Rosinger

Published Dec 14, 2010• 2 citations
OpenAlex

Abstract

A class of non-Cartesian physical systems, [7], are those whose composite state spaces are given by significantly extended tensor products. A more detailed presentation of the way such extended tensor products are constructed is offered, based on a step by step comparison with the construction of usual tensor products. This presentation clarifies the extent to which the extended tensor products are indeed more general than the usual ones. ”History is written with the feet...” Ex-Chairman Mao ”Science in not done scientifically, since it is mostly done by non-scientists... ” Anonymous 1 1. Preliminaries As seen in [7], there is a natural division of physical systems in Cartesian

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A class of non-Cartesian physical systems, [7], are those whose composite state spaces are given by significantly extended tensor products. A more detailed presentation of the way such extended tensor products are constructed is offered, based on a step by step comparison with the construction of usual tensor products. This presentation clarifies the extent to which the extended tensor products are indeed more general than the usual ones. ”History is written with the feet...” Ex-Chairman Mao ”Science in not done scientifically, since it is mostly done by non-scientists... ” Anonymous 1 1. Preliminaries As seen in [7], there is a natural division of physical systems in Cartesian
A recent general model of entanglement, [5], that goes much beyond the usual one based on tensor products of vector spaces is further developed here. It is shown that the usual Cartesian product can be seen as two extreme particular instances of non-entanglement. Also the recent approach to entanglement in [8] is incorporated in the general model in [5]. The idea pursued is that entanglement is by now far too important a phenomenon in Quantum Mechanics, in order to be left confined to its present exclusive modelling by tensor products. Once this is realized, it turns out that one can quite easily de-entangle entanglement from tensor products, and in fact, one can do so in a large variety of ways. Within such general settings the issue of entanglement becomes connected with the issue of "System versus Subsystem" in General Systems Theory, where synthesizing subsystems into a system may often be less difficult than identifying subsystems in a system.
In this work we present an intuitive construction of the quantum logical axiomatic system provided by George Mackey. The goal of this work is a detailed discussion of the results from the paper 'Physical justification for using the tensor product to describe two quantum systems as one joint system' [1] published by Diederik Aerts and Ingrid Daubechies. This means that we want to show how certain composed physical systems from classical and quantum mechanics should be described logically. To reach this goal, we will, like in [1], discuss a special class of axiomatically defined composed physical systems. With the help of certain results from lattice and c-morphism theory (see [2] and [23]), we will present a detailed proof of the statement, that in the quantum mechanical case, a composed physical system must be described via a tensor product space.
Motivated by the novel applications of the mathematical formalism of quantum theory and its generalizations in cognitive science, psychology, social and political sciences, and economics, we extend the notion of the tensor product and entanglement. We also study the relation between conventional entanglement of complex qubits and our generalized entanglement. Our construction can also be used to describe entanglement in the framework of non-Archimedean physics. It is also possible to construct tensor products of non-Archimedean (e.g., p-adic) and complex Hilbert spaces.
Entanglement is a well known fundamental resource in quantum information. Here the following question is addressed : which are the deeper roots of entanglement that may help in its better understanding and use ? The answer is that one can reproduce the phenomenon of entanglement in a far more general and simple way, a way that goes much beyond the usual one which is limited to the framework of tensor products of vector spaces. In this general approach to entanglement presented here - and much unlike in the particular setup of tensor products of vector spaces - the spaces involved can be rather arbitrary sets, just as in the case of Cartesian products. In particular, they need not even have any algebraic structure. Thus they do not have to be vector spaces, groups or even semigroups.
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 orthocomplemented 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 towards this achievement.