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Lecture notes in computer science

Positive Operator-Valued Measures in Quantum Decision Theory

Vyacheslav I. Yukalov, Didier Sornette

Published Jan 1, 2015• 11 citations
Publisher
DOI
10.1007/978-3-319-15931-7_12

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The authors present the theory of quantum measurements in a humanities friendly way. The most general process of decision-making is represented with the aid of the formalism of quantum apparatuses and instruments. This measurement formalism generalizes the standard one based on the von Neumann–Lüders projection postulate. Generalized quantum observables are mathematically represented as positive operator valued measures (POVMs) and state transformers resulting from the feedback of measurements to the states of systems that are given by quantum instruments. The quantum scheme of indirect measurements (a special realization of quantum instruments) is applied to model decision-making as resulting from the interaction between the belief and decision states. The authors analyze the specific features of quantum instruments which are important for cognitive and social applications. In particular, the state transformers given by quantum instruments are in general less invasive than the state projections. Thus quantum-like decision-making need not be viewed as a kind of state collapse. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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We show that the correct mathematical foundation of quantum decision theory, dealing with uncertain events, requires the use of positive operator-valued measure that is a generalization of the projection-valued measure. The latter is appropriate for operationally testable events, while the former is necessary for characterizing operationally uncertain events. In decision making, one has to distinguish composite non-entangled events from composite entangled events. The mathematical definition of entangled prospects is based on the theory of Hilbert-Schmidt spaces and is analogous to the definition of entangled statistical operators in quantum information theory. We demonstrate that 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. The origin of uncertainties in standard lotteries is explained.
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.
The authors present the theory of quantum measurements in a humanities friendly way. The most general process of decision-making is represented with the aid of the formalism of quantum apparatuses and instruments. This measurement formalism generalizes the standard one based on the von Neumann–Lüders projection postulate. Generalized quantum observables are mathematically represented as positive operator valued measures (POVMs) and state transformers resulting from the feedback of measurements to the states of systems that are given by quantum instruments. The quantum scheme of indirect measurements (a special realization of quantum instruments) is applied to model decision-making as resulting from the interaction between the belief and decision states. The authors analyze the specific features of quantum instruments which are important for cognitive and social applications. In particular, the state transformers given by quantum instruments are in general less invasive than the state projections. Thus quantum-like decision-making need not be viewed as a kind of state collapse. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.