Because of the efficiency of modeling fuzziness and vagueness, Z-number plays an important role in real practice. However, Z-numbers, defined in the real number field, lack the ability to process the quantum information in quantum environment. It is reasonable to generalize Z-number into its quantum counterpart. In this paper, we propose quantum Z-numbers (QZNs), which are the quantum generalization of Z-numbers. In addition, seven basic quantum fuzzy operations of QZNs and their corresponding quantum circuits are presented and illustrated by numerical examples. Moreover, based on QZNs, a novel quantum multi-attributes decision making (MADM) algorithm is proposed and applied in medical diagnosis. The results show that, with the help of quantum computation, the proposed algorithm can make diagnoses correctly and efficiently.Because of the efficiency of modeling fuzziness and vagueness, Z-number plays an important role in real practice. However, Z-numbers, defined in the real number field, lack the ability to process the quantum information in quantum environment. It is reasonable to generalize Z-number into its quantum counterpart. In this paper, we propose quantum Z-numbers (QZNs), which are the quantum generalization of Z-numbers. In addition, seven basic quantum fuzzy operations of QZNs and their corresponding quantum circuits are presented and illustrated by numerical examples. Moreover, based on QZNs, a novel quantum multi-attributes decision making (MADM) algorithm is proposed and applied in medical diagnosis. The results show that, with the help of quantum computation, the proposed algorithm can make diagnoses correctly and efficiently.

arXiv:2104.05190Apr 12, 20210 citations

Quantum Numbers

Klaus Hentschel

Compendium of Quantum PhysicsJan 1, 20090 citations

Complex Function Based on the Analysis of Quantum

Cheng-Sheng Liang

This article in accordance with the principle law of elements,in complex variable function construction of periodic structure's power function and logarithmic function,has given Principle Quantum Number n,Angular Quantum Number l,Magnetic Quantum Number m and Spin Quantum Number ms,as well as the relationships between them.Amendments to the old quantum theory of the quantum rules,make conclusions and experimental results and quantum mechanics consistent.Without the introduction of magnetic charges,the magnetic flux quantum rules are given,The establishment of magnetic flux quantum and the Principle Quantum Number n,Angular Quantum Number l,Magnetic Quantum Number m and Spin Quantum Number ms relationship are the result.This article in accordance with the principle law of elements,in complex variable function construction of periodic structure's power function and logarithmic function,has given Principle Quantum Number n,Angular Quantum Number l,Magnetic Quantum Number m and Spin Quantum Number ms,as well as the relationships between them.Amendments to the old quantum theory of the quantum rules,make conclusions and experimental results and quantum mechanics consistent.Without the introduction of magnetic charges,the magnetic flux quantum rules are given,The establishment of magnetic flux quantum and the Principle Quantum Number n,Angular Quantum Number l,Magnetic Quantum Number m and Spin Quantum Number ms relationship are the result.

Journal of Henan Normal UniversityJan 1, 20090 citations

Quantum Numbers

Galina A. Nemova

SPIE eBooksMar 23, 20220 citations

A quantum number theory

Lucas Daiha, Roberto Rivelino

We employ an algebraic procedure based on quantum mechanics to propose a `quantum number theory' (QNT) as a possible extension of the `classical number theory'. We built our QNT by defining pure quantum number operators ( $q$ -numbers) of a Hilbert space that generate classical numbers ( $c$ -numbers) belonging to discrete Euclidean spaces. To start with this formalism, we define a 2-component natural $q$ -number $\textbf{N}$ , such that $\mathbf{N}^{2} \equiv N_{1}^{2} + N_{2}^{2}$ , satisfying a Heisenberg-Dirac algebra, which allows to generate a set of natural $c$ -numbers $n \in \mathbb{N}$ . A probabilistic interpretation of QNT is then inferred from this representation. Furthermore, we define a 3-component integer $q$ -number $\textbf{Z}$ , such that $\mathbf{Z}^{2} \equiv Z_{1}^{2} + Z_{2}^{2} + Z_{3}^{2}$ and obeys a Lie algebra structure. The eigenvalues of each $\textbf{Z}$ component generate a set of classical integers $m \in \mathbb{Z}\cup \frac{1}{2}\mathbb{Z}^{*}$ , $\mathbb{Z}^{*} = \mathbb{Z} \setminus \{0\}$ , albeit all components do not generate $\mathbb{Z}^3$ simultaneously. We interpret the eigenvectors of the $q$ -numbers as ` $q$ -number state vectors' (QNSV), which form multidimensional orthonormal basis sets useful to describe state-vector superpositions defined here as qu $n$ its. To interconnect QNSV of different dimensions, associated to the same $c$ -number, we propose a quantum mapping operation to relate distinct Hilbert subspaces, and its structure can generate a subset $W \subseteq \mathbb{Q}^{*}$ , the field of non-zero rationals. In the present description, QNT is related to quantum computing theory and allows dealing with nontrivial computations in high dimensions.We employ an algebraic procedure based on quantum mechanics to propose a `quantum number theory' (QNT) as a possible extension of the `classical number theory'. We built our QNT by defining pure quantum number operators ($q$-numbers) of a Hilbert space that generate classical numbers ($c$-numbers) belonging to discrete Euclidean spaces. To start with this formalism, we define a 2-component natural $q$-number $\textbf{N}$, such that $\mathbf{N}^{2} \equiv N_{1}^{2} + N_{2}^{2}$, satisfying a Heisenberg-Dirac algebra, which allows to generate a set of natural $c$-numbers $n \in \mathbb{N}$. A probabilistic interpretation of QNT is then inferred from this representation. Furthermore, we define a 3-component integer $q$-number $\textbf{Z}$, such that $\mathbf{Z}^{2} \equiv Z_{1}^{2} + Z_{2}^{2} + Z_{3}^{2}$ and obeys a Lie algebra structure. The eigenvalues of each $\textbf{Z}$ component generate a set of classical integers $m \in \mathbb{Z}\cup \frac{1}{2}\mathbb{Z}^{*}$, $\mathbb{Z}^{*} = \mathbb{Z} \setminus \{0\}$, albeit all components do not generate $\mathbb{Z}^3$ simultaneously. We interpret the eigenvectors of the $q$-numbers as `$q$-number state vectors' (QNSV), which form multidimensional orthonormal basis sets useful to describe state-vector superpositions defined here as qu$n$its. To interconnect QNSV of different dimensions, associated to the same $c$-number, we propose a quantum mapping operation to relate distinct Hilbert subspaces, and its structure can generate a subset $W \subseteq \mathbb{Q}^{*}$, the field of non-zero rationals. In the present description, QNT is related to quantum computing theory and allows dealing with nontrivial computations in high dimensions.

arXiv:2108.10145Aug 18, 20210 citations

Quantum Numbers and Rules

Paul Peter Urone, Roger A. Hinrichs

Jan 1, 20120 citations

q

-numbers) of a Hilbert space that generate classical numbers (

c

-numbers) belonging to discrete Euclidean spaces. To start with this formalism, we define a 2-component natural

q

-number

\textbf{N}

, such that

\mathbf{N}^{2} \equiv N_{1}^{2} + N_{2}^{2}

, satisfying a Heisenberg-Dirac algebra, which allows to generate a set of natural

c

-numbers

n \in \mathbb{N}

. A probabilistic interpretation of QNT is then inferred from this representation. Furthermore, we define a 3-component integer

q

-number

\textbf{Z}

, such that

\mathbf{Z}^{2} \equiv Z_{1}^{2} + Z_{2}^{2} + Z_{3}^{2}

and obeys a Lie algebra structure. The eigenvalues of each

\textbf{Z}

component generate a set of classical integers

m \in \mathbb{Z}\cup \frac{1}{2}\mathbb{Z}^{*}

\mathbb{Z}^{*} = \mathbb{Z} \setminus \{0\}

, albeit all components do not generate

\mathbb{Z}^3

simultaneously. We interpret the eigenvectors of the

q

-numbers as `

q

-number state vectors' (QNSV), which form multidimensional orthonormal basis sets useful to describe state-vector superpositions defined here as qu

n

its. To interconnect QNSV of different dimensions, associated to the same

c

-number, we propose a quantum mapping operation to relate distinct Hilbert subspaces, and its structure can generate a subset

W \subseteq \mathbb{Q}^{*}

, the field of non-zero rationals. In the present description, QNT is related to quantum computing theory and allows dealing with nontrivial computations in high dimensions.