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Applications of Mathematics

A new approach to representation of observables on fuzzy quantum posets

Lê Bá Long

Published Jan 1, 1992• 6 citations
Publisher

Abstract

We give a representation of an observable on a fuzzy quantum poset of type II by a pointwise defined real-valued function. This method is inspired by that of Kolesárová [6] and Mesiar [7], and our results extend representations given by the author and Dvurečenskij [4]. Moreover, we show that in this model, the converse representation fails, in general.

DOI
10.21136/am.1992.104516

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We give a representation of an observable on a fuzzy quantum poset of type II by a pointwise defined real-valued function. This method is inspired by that of Kolesárová [6] and Mesiar [7], and our results extend representations given by the author and Dvurečenskij [4]. Moreover, we show that in this model, the converse representation fails, in general.
A new probability model based on the theory of fuzzy sets is presented. In thismodel, a difference of comparable fuzzy sets is the primary operation. The idea ofa difference of fuzzy sets (fuzzy events) is simple: If we have two comparable eventsa and b ( a ≤ b ), then our knowledge on a and b entails the complete knowledge ofthe complement of a in b , i. e., b ⊖ a .The new algebraic structure of fuzzy sets is called a difference poset (a D-poset) offuzzy sets. Some properties of a lattice ordered D-poset of fuzzy sets (a D-lattice offuzzy sets) are studied. An MV-algebra of fuzzy sets (a Bold algebra) is characterizedin the D-poset of fuzzy sets set-up. The sufficient and necessary conditions for a Dlatticeof fuzzy sets to be a Bold algebra are given. The basic notions of the quantumlogic theory - a state and an observable are defined in D-posets of fuzzy sets.
In this paper, we propose a new interpretation of quantum mechanics according to which Born probabilities are interpreted as many-valued truth values of statements concerning results of not-yet-performed experiments on quantum objects, or equivalently [0,1]-valued degrees to which quantum objects possess all their properties prior to measurements. The interpretation is based on mathematical results already published by the author, according to which any Birkhoff–von Neumann quantum logic can be isomorphically represented as a family of fuzzy sets or, equivalently, many-valued propositional functions.
In this article we investigate a way in which quantum computing can be used to extend the class of fuzzy sets. The core idea is to see states of a quantum register as characteristic functions of quantum fuzzy subsets of a given set. As the real unit interval is embedded in the Bloch sphere, every fuzzy set is automatically a quantum fuzzy set. However, a generic quantum fuzzy set can be seen as a (possibly entangled) superposition of many fuzzy sets at once, offering new opportunities for modeling uncertainty. After introducing the main framework of quantum fuzzy set theory, we analyze the standard operations of fuzzification and defuzzification from our viewpoint. We conclude this preliminary paper with a list of possible applications of quantum fuzzy sets to pattern recognition, as well as future directions of pure research in quantum fuzzy set theory.