Quantum Measurement Trees, I: Two Preliminary Examples of Induced Contextual Boolean Algebras

Abstract

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 due to Feynman (1951, 1966). A related non-quantum example, inspired by Boole (1862) and Vorob$'$ev (1962), 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 $-algebras of measurable events. This concept due to Vorob$'$ev (1962) allows construction of: 1) a measurable meta\-space whose elements combine a point in the original sample space with a variable ``contextual'' Boolean algebra; 2) a parametric family of probability meta\-spaces, 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.