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ON SOME CONTRIBUTIONS TO QUANTUM STRUCTURES BY FUZZY SETS

Beloslav Rie

Published Jan 1, 2007• 2 citations
OpenAlex

Abstract

It is well known that the fuzzy sets theory can be successfully used in quantum models ([5,26]). In this paper we give first a review of recent development in the probability theory on tribes and their generalizations ‐ multivalued (MV)-algebras. Secondly we show some applications of the described method to develop probability theory on IF-events.

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Cambridge University Press eBooks•Apr 2, 1998•0 citations
It is well known that the fuzzy sets theory can be successfully used in quantum models ([5,26]). In this paper we give first a review of recent development in the probability theory on tribes and their generalizations ‐ multivalued (MV)-algebras. Secondly we show some applications of the described method to develop probability theory on IF-events.
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.
Quantum computational logics provide a fertile common ground for a unied treatment of vagueness and uncertainty. In this paper we describe an approach to the logic of quantum computation that has been recently taken up and developed by the present authors. Special attention will be devoted to a generalisation of Chang's MV algebras (called quasi-MV alge- bra) which abstracts over the algebra whose universe is the set of qumixes of the 2-dimensional complex Hilbert space, as well as to its expansions by additional quantum connectives. We fur- thermore explore some future research perspectives, also in the light of some recent limitative results whose general signicance will be duly assessed.
This paper surveys some of the recent results that have been obtained in fuzzy quantum probability theory. In this theory fuzzy events are represented by quantum effects and fuzzy random variables are represented by quantum measurements. Probabilities, conditional probabilities and sequential products of effects are discussed. The relationship between the law of total probability and compatibility of measurements is treated. Results concerning independent effects are given. Finally, properties of the sequential product, almost sharp effects and nearly sharp effects are discussed.
Quantum probability and quantum mechanics The formalism of quantum probability theory Discussions of quantum mechanics are often confused by a lack of clarity about what exactly constitutes ‘quantum mechanics’. It is therefore useful to try at the start to isolate a consistent mathematical core of quantum mechanics, and consider anything that goes beyond this core to be ‘interpretation’. For us, this core is quantum probability theory. Quantum probability is a generalization of classical probability, and therefore I begin with a brief review of the latter. I assume that the reader has some familiarity with the ideas of probability theory. What follows is just to provide a quick review, and to establish some notation and terminology. In modern classical probability theory, probabilities are defined over algebras of events. The motivation is straightforward: we begin with a set of ‘primitive’, or ‘simple’, events (the ‘sample space’), and form an algebra of events by taking all logical combinations of the simple events. For example, let us take the simple events to be the possible results of rolling a six-sided die one time, so that the sample space is the set {1,2,3,4,5,6}. We then form an algebra of events from the sample space by taking all possible logical combinations of the simple events. Logical combinations include, for example, ‘either 3 or 5’ and ‘not 3 and not 2’.