The Mathematical Framework of Quantum Theory

Trust models play an important role in computational environments. One of the main aims of the work undertaken in this domain is to provide a model that can better describe the socio-technical nature of computational trust. It has been recently shown that quantum-like formulations in the field of human decision making can better explain the underlying nature of these types of processes. Based on this research, the aim of this paper is to propose a novel model of trust based on quantum probabilities as the underlying mathematics of quantum theory. It will be shown that by using this new mathematical framework, we will have a powerful mechanism to model the contextuality property of trust. Also, it is hypothesized that many events or evaluations in the context of trust can be and should be considered as incompatible, which is unique to the noncommutative structure of quantum probabilities. The main contribution of this paper will be that, by using the quantum Bayesian inference mechanism for belief updating in the framework of quantum theory, we propose a biased trust inference mechanism. This mechanism allows us to model the negative and positive biases that a trustor may subjectively feel toward a certain trustee candidate. It is shown that by using this bias, we can model and describe the exploration versus exploitation problem in the context of trust decision making, recency effects for recently good or bad transactions, filtering pessimistic and optimistic recommendations that may result in good-mouthing or bad-mouthing attacks, the attitude of the trustor toward risk and uncertainty in different situations and the pseudo-transitivity property of trust. Finally, we have conducted several experimental evaluations in order to demonstrate the effectiveness of the proposed model in different scenarios.

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The Mathematical Framework of Quantum Theory

The Physics of Atoms and Quanta ◽

10.1007/978-3-642-97468-7_9 ◽

1993 ◽

pp. 119-146

Author(s):

Hermann Haken ◽

Hans Christoph Wolf

Keyword(s):

Quantum Theory ◽

Mathematical Framework

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Probability in Hilbert Space

Quantum Theory ◽

10.1093/oso/9780192896308.003.0003 ◽

2021 ◽

pp. 93-167

Author(s):

Jochen Rau

Keyword(s):

Hilbert Space ◽

Quantum Theory ◽

Experimental Evidence ◽

Linear Algebra ◽

Classical Limit ◽

Logical Structure ◽

Complex Hilbert Space ◽

Mathematical Representation ◽

Mathematical Framework ◽

Two Level Systems

This chapter introduces the mathematical framework, basic rules, and some key results of quantum theory. After a succinct overview of linear algebra and an introduction to complex Hilbert space, it investigates the correspondence between subspaces of Hilbert space and propositions, their logical structure, and how the pertinent probabilities are calculated. It discusses the mathematical representation of states, observables, and transformations, as well as the rules for calculating expectation values and uncertainties, and for updating states after a measurement. Particular attention is paid to two-level systems, or ‘qubits’, and the connection is made with experimental evidence about binary measurements. The properties of composite systems are discussed in detail, notably the phenomenon of entanglement. The chapter concludes with an investigation of conceptual issues regarding realism, non-contextuality, and locality, as well as the classical limit.

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