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A note on quantum entanglement in Dempster-Shafer evidence theory

Xinyang Deng, Yong Deng

Published Mar 1, 2015• 0 citations
OpenAlex

Abstract

Dempster-Shafer evidence theory is an efficient mathematical tool to deal with uncertain information. In this theory, basic probability assignment (BPA) is the basic structure for the expression and inference of uncertainty. In this paper, quantum entanglement involved in Dempster-Shafer evidence theory is studied. A criterion is given to determine whether a BPA is in an entangled state or not. Based on that, the information volume involved in a BPA is discussed. The discussion shows that a non-quantum strategy (or observation) can not obtained all information contained in a BPA which is in an entangled state.

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Dempster-Shafer evidence theory is an efficient mathematical tool to deal with uncertain information. In this theory, basic probability assignment (BPA) is the basic structure for the expression and inference of uncertainty. In this paper, quantum entanglement involved in Dempster-Shafer evidence theory is studied. A criterion is given to determine whether a BPA is in an entangled state or not. Based on that, the information volume involved in a BPA is discussed. The discussion shows that a non-quantum strategy (or observation) can not obtained all information contained in a BPA which is in an entangled state.
Theory of quantum measurements is often classified as decision theory. An event in decision theory corresponds to the measurement of an observable. This analogy looks clear for operationally testable simple events. However, the situation is essentially more complicated in the case of composite events. The most difficult point is the relation between decisions under uncertainty and measurements under uncertainty. We suggest a unified language for describing the processes of quantum decision making and quantum measurements. The notion of quantum measurements under uncertainty is introduced. We show that the correct mathematical foundation for the theory of measurements under uncertainty, as well as for quantum decision theory dealing with uncertain events, requires the use of positive operator-valued measure that is a generalization of projection-valued measure. The latter is appropriate for operationally testable events, while the former is necessary for characterizing operationally uncertain events. In both decision making and quantum measurements, one has to distinguish composite nonentangled events from composite entangled events. Quantum probability can be essentially different from classical probability only for entangled events. The necessary condition for the appearance of an interference term in the quantum probability is the occurrence of entangled prospects and the existence of an entangled strategic state of a decision maker or of an entangled statistical state of a measuring device.
Quantum entanglement, a cornerstone of quantum mechanics, presents a profoundly perplexing phenomenon. Its implications extend far beyond basic quantum theory, offering the potential for revolutionary advancements in computation and communication. This paper explores a novel theoretical framework, the Quantum Entanglement Probability Expansion (QEP), designed to significantly deepen our understanding of the relationship between probability, system complexity, and the efficiency of quantum information processing. We propose a modified probabilistic model that explicitly incorporates the entanglement properties of quantum systems, allowing for a more precise and computationally efficient approach to quantum information handling. The core mechanism centers around extending the probability distribution of quantum states to encompass the entire system, enabling a holistic assessment of information processing dynamics. This paper details the QEP's mathematical foundations, its potential applications within quantum information theory, and preliminary results demonstrating increased efficiency compared to conventional approaches. The implications of this research extend beyond mere theoretical advancements, potentially paving the way for practical quantum algorithms and enhanced quantum communication protocols.
The sure thing principle and the law of total probability are basic laws in classic probability theory. A disjunction fallacy leads to the violation of these two classical probability laws. In this paper, a new quantum dynamic belief decision making model based on quantum dynamic modelling and Dempster-Shafer (D-S) evidence theory is proposed to address this issue and model the real human decision-making process. Some mathematical techniques are borrowed from quantum mathematics. Generally, belief and action are two parts in a decision making process. The uncertainty in belief part is represented by a superposition of certain states. The uncertainty in actions is represented as an extra uncertainty state. The interference effect is produced due to the entanglement between beliefs and actions. Basic probability assignment (BPA) of decisions is generated by quantum dynamic modelling. Then BPA of the extra uncertain state and an entanglement degree defined by an entropy function named Deng entropy are used to measure the interference effect. Compared the existing model, the number of free parameters is less in our model. Finally, a classical categorization decision-making experiment is illustrated to show the effectiveness of our model.
We point out that density matrices can only be used to describe quantum states, so the entanglement contained in a density matrix is just quantum entanglement. This means a bipartite state described by a density matrix contains quantum entanglement, unless the density matrix has the form ρAB=ρA⊗ρBρ_{AB}=ρ_A\otimes ρ_BρAB​=ρA​⊗ρB​.
We propose a new interpretation of Dempster-Shafer theory of evidence based on a stochastic model for a quantum system developed in terms of amplitude on entities. An entity provides an axiomatic description for the set of tests and states of a physical system and an amplitude density gives a mean for computing probability and pseudo-probability within this framework. The interference and independence of tests relative to an amplitude density are formulated. Various ways of combining entities and amplitudes are presented as superpositions of amplitude. We propose to extend the theory of evidence also to the physical observable experiment in quantum mechanics. So we can use, for quantum mechanics, all the knowledge in the evidence theory. On the contrary the knowledge in the evidence theory can give new insight in the quantum mechanics uncertainty. We give a modal logic interpretation of evidence theory and entity and tests definitions. We establish a bridge between mental approach to uncertainty by evidence theory and the physical phenomena represented by quantum mechanics. When one use the amplitude density in the evidence theory one can generate a new interpretation of joint and conditional definition of the basic probability assignment ( pseudo-probability).