Fuzzy model theory is an alternative approach to formalizing uncertainty and/or incompleteness of knowledge to the formalization using fuzzy sets proposed by Lotfi Zadeh. This approach to information processing is also applied in quantum computing. All quantum computations are conducted under the assumption of incomplete knowledge and only at the last stage the measurement (i.e. digitization) of the obtained results is performed. The paper describes a model-theoretic formalization of a pure quantum system in the language of fuzzy model theory. It is shown that the fuzzy model can be considered as a generalization of the concept of quantum system. One of the central concepts of quantum computing is quantum entanglement. A quantum system is considered to be non-entangled or separable if the results of measurements of some elements of the system do not depend on measurements of other elements. The paper introduces the concept of entangled and partially entangled fuzzy system, formulates and proves the criteria of entanglement of a fuzzy system.
Fuzzy model theory is an alternative approach to formalizing uncertainty and/or incompleteness of knowledge to the formalization using fuzzy sets proposed by Lotfi Zadeh. This approach to information processing is also applied in quantum computing. All quantum computations are conducted under the assumption of incomplete knowledge and only at the last stage the measurement (i.e. digitization) of the obtained results is performed. The paper describes a model-theoretic formalization of a pure quantum system in the language of fuzzy model theory. It is shown that the fuzzy model can be considered as a generalization of the concept of quantum system. One of the central concepts of quantum computing is quantum entanglement. A quantum system is considered to be non-entangled or separable if the results of measurements of some elements of the system do not depend on measurements of other elements. The paper introduces the concept of entangled and partially entangled fuzzy system, formulates and proves the criteria of entanglement of a fuzzy system.
One of the directions of application of the theory of fuzzy models is model-theoretic formalization of quantum computing. The central concept of quantum computing is quantum entanglement. A quantum system is considered to be unentangled or separable if the results of measurements of some elements of the system do not depend on measurements of other elements.Fuzzy Model can be considered as a generalization of quantum system. The paper deals with the generalization of the separability property of the system to the case of arbitrary Fuzzy Model. The definition of Fuzzy Separable Model is given. It is proved that under given initial conditions (atomic diagram of the model) there exists not more than one Separable Fuzzy Model. The criterion, which the distribution function of a fuzzy model must have in order for this model to become separable, is described. The notion of the rank of the Separable Model is introduced, which can be regarded as a measure of entropy of fuzziness of the Separable Model.
Recently we proposed the linguistic Copenhagen interpretation of quantum mechanics, which is called quantum language or measurement theory. This theory is valid for both quantum and classical systems. Thus we think that quantum language is one of the most powerful scientific theories, like statistics. In this paper we justify Zadeh’s fuzzy sets theory in quantum language, that is, fuzzy propositions are identified with binary measurements. This implies that the definition of “proposition” is, for the first time, acquired in the field of non-mathematics. Further, we assert that fuzzy logic is more natural than crisp logic in science. And furthermore, we discuss and solve Saussure’s linguistics, Moore’s paradox, Quine’s analytic-synthetic distinction and Lewis Carroll’s logical paradox. Therefore, from the philosophical point of view, our result gives a complete answer to Wittgenstein’s problem: “Why does logic work in our world?” and “What is a scientific proposition?” in his picture theory. That is, we simultaneously justify both Zadeh’s fuzzy sets and Wittgenstein’s picture theory in the quantum mechanical worldview.
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.
Quantum entangled states play very important roles in quantum information processing,such as quantum teleportation,dense coding,error correction,cryptography,and computation.The theory of entanglement mainly concerns the characterization of quantum entanglement and the classification of quantum states,as well as its applications in quantum information processing.We present some basic concepts and results,including:the relationship between locality and reality in quantum mechanics and Bell inequalities,and that between Bell inequalities and the separability of quantum states;the definition of separability for pure and mixed quantum states,and some separability criteria such as the positive map approach,positive partial transpose,reduction criterion,realignment,entanglement witnesses,and the covariance matrix approach and local uncertainty relations;some entanglement measures such as the entanglement of formation,concurrence,relative entropy,negativity,tangle,entanglement of assistance,and their estimation;the change in entanglement when the physical system evolves or undergoes interaction with the environment.
In this paper, the formulation of Quantum Mechanics in terms of fuzzy logic and fuzzy sets is explored. A result by Pykacz, which establishes a correspondence between (quantum) logics (lattices with certain properties) and certain families of fuzzy sets, is applied to the Birkhoff–von Neumann logic, the lattice of projectors of a Hilbert space. Three cases are considered: the qubit, two qubits entangled, and a qutrit ‘nested’ inside the two entangled qubits. The membership functions of the fuzzy sets are explicitly computed and all the connectives of the fuzzy sets are interpreted as operations with these particular membership functions. In this way, a complete picture of the standard quantum logic in terms of fuzzy sets is obtained for the systems considered.
In this paper we consider physical systems and the concept of their states in the context of the theory of fuzzy sets and systems. In section 1 we give a brief sketch on the fundamental difference between the theories of classical physics and quantum mechanics. In section 2 and 3 we introduce very shortly systems and their states in classical and quantum mechanics, respectively. Section 4 presents the concept of fuzzy systems. We propose to fuzzify the classical systems in section 5 and quantum systems in section 6. In section 7 we start to consider a fuzzy interpretation of the uncertainty principle.