Quantum dynamical mode (QDM): A possible extension of belief function

TL;DR

This paper introduces a generalized Dempster-Shafer evidence theory using complex numbers, enabling integration with quantum models and extending applicability beyond classical limitations, demonstrated through numerical examples and an evidential quantum dynamical application.

Contribution

It proposes a complex-valued mass function and a generalized combination rule, bridging classical evidence theory with quantum theory and relaxing previous conflict constraints.

Findings

The generalized theory is more flexible than classical evidence theory.

Numerical examples demonstrate the efficiency of the generalized approach.

An application shows the feasibility of integrating evidence theory with quantum models.

Abstract

Dempster-Shafer evidence theory has been widely used in various fields of applications, because of the flexibility and effectiveness in modeling uncertainties without prior information. Besides, it has been proven that the quantum theory has powerful capabilities of solving the decision making problems, especially for modelling human decision and cognition. However, the classical Dempster-Shafer evidence theory modelled by real numbers cannot be integrated directly with the quantum theory modelled by complex numbers. So, how can we establish a bridge of communications between the classical Dempster-Shafer evidence theory and the quantum theory? To answer this question, a generalized Dempster-Shafer evidence theory is proposed in this paper. The main contribution in this study is that, unlike the existing evidence theory, a mass function in the generalized Dempster-Shafer evidence theory is modelled by a complex number, called as a complex mass function. In addition, compared with the classical Dempster's combination rule, the condition in terms of the conflict coefficient between two evidences K < 1 is released in the generalized Dempster's combination rule so that it is more general and applicable than the classical Dempster's combination rule. When the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster's combination rule degenerates to the classical evidence theory under the condition that the conflict coefficient between the evidences K is less than 1. Numerical examples are illustrated to show the efficiency of the generalized Dempster-Shafer evidence theory. Finally, an application of an evidential quantum dynamical model is implemented by integrating the generalized Dempster-Shafer evidence theory with the quantum dynamical model. From the experimental results, it validates the feasibility and effectiveness of the proposed method.