On a possibility to combine the order effect with sequential reproducibility for quantum measurements

TL;DR

This paper investigates whether quantum observables can simultaneously exhibit the order effect and sequential reproducibility, revealing that certain types of reproducibility can coexist with the order effect, but not all.

Contribution

It demonstrates that for POVMs, the order effect can coexist with some forms of reproducibility, but not with the combined separated reproducibility of the $B-A-B$ type.

Findings

Order effect can coexist with adjacent and separated reproducibility for POVMs.

Separated reproducibility of the $B-A-B$ type prevents the coexistence with the order effect.

Results are relevant for quantum foundations and applications in psychology and psychophysics.

Abstract

In this paper we study the problem of a possibility to use quantum observables to describe a possible combination of the order effect with sequential reproducibility for quantum measurements. By the order effect we mean a dependence of probability distributions (of measurement results) on the order of measurements. We consider two types of the sequential reproducibility: adjacent reproducibility ($A-A$) and separated reproducibility($A-B-A$). The first one is reproducibility with probability 1 of a result of measurement of some observable $A$ measured twice, one $A$ measurement after the other. The second one, $A-B-A$, is reproducibility with probability 1 of a result of $A$ measurement when another quantum observable $B$ is measured between two $A$'s. Heuristically, it is clear that the second type of reproducibility is complementary to the order effect. We show that, surprisingly, for an important class of quantum observables given by positive operator valued measures (POVMs), this may not be the case. The order effect can coexist with a separated reproducibility as well as adjacent reproducibility for both observables $A$ and $B.$ However, the additional constraint in the form of separated reproducibility of the $B-A-B$ type makes this coexistence impossible. Mathematically, this paper is about the operator algebra for effects composing POVMs. The problem under consideration was motivated by attempts to apply the quantum formalism outside of physics, especially, in cognitive psychology and psychophysics. However, it is also important for foundations of quantum physics as a part of the problem about the structure of sequential quantum measurements.