Quantum measurement trees, I: two preliminary examples of induced contextual Boolean algebras

Quantum randomness evidently transcends the classical framework of random variables defined on a single comprehensive Kolmogorov probability space. One prominent example is the quantum double-slit experiment owing to Feynman (Feynman 1951 In Second Berkeley Symposium on Mathematical Statistics and Probability (ed. J. Neyman), pp. 533-541 (doi:10.1525/9780520411586-039)). A related non-quantum example, inspired by Boole (Boole 1862 Phil. Trans. R. Soc. Lond. 152, 225-252 (doi:10.1098/rstl.1862.0015)) and Vorob'ev (Vorob'ev 1962 Theory Probab. Appl.7, 147-163 (doi:10.1137/1107014)), has three two-valued random variables X, Y and Z, where the pairs X, Y and X, Z are perfectly correlated, yet Y, Z are perfectly anti-correlated. Such examples can be accommodated using a 'multi-measurable' space with several different sigma-algebras of measurable events. This concept, owing to Vorob'ev (Vorob'ev 1962 Theory Probab. Appl.7, 147-163 (doi:10.1137/1107014)), allows construction of (i) a measurable metaspace whose elements combine a point in the original sample space with a variable 'contextual' Boolean algebra; (ii) a parametric family of 'probability metaspaces', each of which is a Kolmogorov probability space that represents a two-stage stochastic process where a random choice from the original sample space is preceded by the random choice of a contextual Boolean algebra in the multi-measurable space. Subsequent work will explore how quantum experimental results can be described using a quantum measurement tree with one or more preparation nodes, where an experimental configuration is determined that governs the probability distribution of relevant quantum observables. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.