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===6.3. “QOE+RRE”: described by quantum instruments of non-projective type=== | ===6.3. “QOE+RRE”: described by quantum instruments of non-projective type=== | ||
In paper (Khrennikov et al., 2014),<ref name=":0" /> it was shown that by using the von Neumann calculus it is ''impossible to combine RRE with QOE.'' To generate QOE, Hermitian operators <math>\widehat{A},\widehat{B} </math> should be noncommutative, but the latter destroys<math>A-B-A </math> response replicability of <math>A </math>. This was a rather unexpected result. It made even impression that, although the basic cognitive effects can be quantum-likely modeled separately, their combinations cannot be described by the quantum formalism. | In paper (Khrennikov et al., 2014),<ref name=":0" /> it was shown that by using the von Neumann calculus it is ''impossible to combine RRE with QOE.'' To generate QOE, Hermitian operators <math>\widehat{A},\widehat{B} </math> should be noncommutative, but the latter destroys<math>A-B-A </math> response replicability of <math>A </math>. This was a rather unexpected result. It made even impression that, although the basic cognitive effects can be quantum-likely modeled separately, their combinations cannot be described by the quantum formalism. | ||
However, recently it was shown that theory of quantum instruments provides a simple solution of the combination of QOE and RRE effects, see Ozawa and Khrennikov (2020a)<ref name=":1">Ozawa M., Khrennikov A. Application of theory of quantum instruments to psychology: Combination of question order effect with response replicability effect. Entropy, 22 (1) (2020</ref> for construction of such instruments. These instruments are of non-projective type. Thus, the essence of QOE is not in the structure of observables, but in the structure of the state transformation generated by measurements’ feedback. QOE is not about the joint measurement and incompatibility (noncommutativity) of observables, but about sequential measurement of observables and sequential (mental-)state update. Quantum instruments which are used in Ozawa and Khrennikov (2020a)<ref name=":1" /> to combine QOE and RRE correspond to measurement of observables represented by commuting operators <math>\widehat{A},\widehat{B} </math>. Moreover, it is possible to prove that (under natural mathematical restriction) QOE and RRE can be jointly modeled only with the aid of quantum instruments for commuting observables. | However, recently it was shown that theory of quantum instruments provides a simple solution of the combination of QOE and RRE effects, see Ozawa and Khrennikov (2020a)<ref name=":1">Ozawa M., Khrennikov A. Application of theory of quantum instruments to psychology: Combination of question order effect with response replicability effect. Entropy, 22 (1) (2020</ref> for construction of such instruments. These instruments are of non-projective type. Thus, the essence of QOE is not in the structure of observables, but in the structure of the state transformation generated by measurements’ feedback. QOE is not about the joint measurement and incompatibility (noncommutativity) of observables, but about sequential measurement of observables and sequential (mental-)state update. Quantum instruments which are used in Ozawa and Khrennikov (2020a)<ref name=":1" /> to combine QOE and RRE correspond to measurement of observables represented by commuting operators <math>\widehat{A},\widehat{B} </math>. Moreover, it is possible to prove that (under natural mathematical restriction) QOE and RRE can be jointly modeled only with the aid of quantum instruments for commuting observables. | ||