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arXiv:2206.05185 · International Journal of Theoretical Physics

Decision-making under uncertainty – a quantum value operator approach

Lizhi Xin, Houwen Xin

Published Feb 28, 2023• 9 citations
arXivHTMLPDF
DOI
10.1007/s10773-023-05308-w

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Abstract Decision’s biggest challenge is uncertainty. We propose a quantum expected value theory for decision-making under uncertainty. Decision-making under uncertainty is the unification of people’s subjective beliefs and the objective world. Quantum density operator as value operator which we call it quantum circuit is proposed to simulate people’s subjective beliefs. Quantum circuit guides people to choose corresponding actions based on their subjective beliefs through objective world. The quantum circuit is a decision tree constructed from quantum gates and logic operations. The genetic programming can be used to optimize and auto-generate quantum circuits. Our proposed synapse’s quantum circuit hypothesis states that the combination of neurotransmitter and receptor functioned as quantum circuit open or close ion channels probabilistically which determines the excitation or inhibition of synapse.
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.