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Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

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

Peter J. Hammond

Published Dec 11, 2025• 1 citation
Publisher

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 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)'.

DOI
10.1098/rsta.2024.0560

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

Peter J. Hammond

This paper is also a CRETA discussion paper series number 90

•Feb 1, 2025•0 citations
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)'.
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 XXX, YYY and ZZZ, where the pairs X,YX, YX,Y and X,ZX, ZX,Z are perfectly correlated, yet Y,ZY, ZY,Z are perfectly anti-correlated. Such examples can be accommodated using a ``multi-measurable'' space with several different $ σ−algebrasofmeasurableevents.ThisconceptduetoVorob-algebras of measurable events. This concept due to Vorob−algebrasofmeasurableevents.ThisconceptduetoVorob'$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.
Given a quantum state in the finite-dimensional Hilbert space Cn, the range of possible values of a quantum observable is usually identified with the discrete spectrum of eigenvalues of a corresponding Hermitian matrix. Here any such observable is identified with (i) an 'ortho-measurable' function defined on the Boolean 'ortho-algebra' generated by the eigenspaces that form an orthogonal decomposition of Cn, and (ii) a 'numerically identified' orthogonal decomposition of Cn. The latter means that each subspace of the orthogonal decomposition can be uniquely identified by its own attached real number, just as each eigenspace of a Hermitian matrix can be uniquely identified by the corresponding eigenvalue. Furthermore, any density matrix on Cn is identified with a Bayesian prior 'ortho-probability' measure defined on the linear subspaces that make up the Boolean ortho-algebra induced by its eigenspaces. Then any pure quantum state is identified with a degenerate density matrix, and any mixed state with a probability measure on a set of orthogonal pure states. Finally, given any quantum observable, the relevant Bayesian posterior probabilities of measured outcomes can be found by the usual trace formula that extends Born's rule. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.
The main aim of this report is to inform the quantum information community about investigations on the problem of probabilistic compatibility of a family of random variables: a possibility to realize such a family on the basis of a single probability measure (to construct a single Kolmogorov probability space). These investigations were started hundred of years ago by J. Boole (who invented Boolean algebras). The complete solution of the problem was obtained by Soviet mathematician Vorobjev in 60th. Surprisingly probabilists and statisticians obtained inequalities for probabilities and correlations among which one can find the famous Bell's inequality and its generalizations. Such inequalities appeared simply as constraints for probabilistic compatibility. In this framework one can not see a priori any link to such problems as nonlocality and "death of reality" which are typically linked to Bell's type inequalities in physical literature. We analyze the difference between positions of mathematicians and quantum physicists. In particular, we found that one of the most reasonable explanations of probabilistic incompatibility is mixing in Bell's type inequalities statistical data from a number of experiments performed under different experimental contexts.
In quantum physics, the term ‘contextual’ can be used in more than one way. One usage, here called ‘Bell contextual’ since the idea goes back to Bell, is that if A , B and C are three quantum observables, with A compatible (i.e. commuting) with B and also with C , whereas B and C are incompatible, a measurement of A might yield a different result (indicating that quantum mechanics is contextual) depending upon whether A is measured along with B (the { A , B } context) or with C (the { A , C } context). An analysis of what projective quantum measurements measure shows that quantum theory is Bell non-contextual: the outcome of a particular A measurement when A is measured along with B would have been exactly the same if A had, instead, been measured along with C . A different definition, here called ‘globally (non)contextual’ refers to whether or not there is (non-contextual) or is not (contextual) a single joint probability distribution that simultaneously assigns probabilities in a consistent manner to the outcomes of measurements of a certain collection of observables, not all of which are compatible. A simple example shows that such a joint probability distribution can exist even in a situation where the measurement probabilities cannot refer to properties of a quantum system, and hence lack physical significance, even though mathematically well defined. It is noted that the quantum sample space, a projective decomposition of the identity, required for interpreting measurements of incompatible properties in different runs of an experiment using different types of apparatus, has a tensor product structure, a fact sometimes overlooked. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.
Since 1933, when Kolmogorov laid the foundations for probability and statistics as we know them today [1], it has been recognized that propositions asserting that such and such an event occurred as a consequence of the execution of a particular random experiment tend to band together and form a Boolean algebra. In 1936, Birkhoff and von Neumann [2] suggested that the so-called logic of quantum mechanics should not be a Boolean algebra, but rather should form what is now called a modular ortholattice [3]. Presumably, the departure from Boolean algebras encountered in quantum mechanics can be attributed to the fact that in quantum mechanics, one must consider more than one physical experiment, e.g., an experiment measuring position, an experiment measuring charge, an experiment measuring momentum, etc., and, because of the uncertainty principle, these experiments need not admit a common refinement in terms of which the Kolmogorov theory is directly applicable. Mackey's Axioms I–VI for quantum mechanics [4] imply that the logic of quantum mechanics should be a σ-orthocomplete orthomodular poset [5]. Most contemporary practitioners of quantum logic seem to agree that a quantum logic is (at least) an orthomodular poset [6], [7], [8], [9], [10] or some variation thereof [11]. P. D. Finch [12] has shown that every completely orthomodular poset is the logic arising from sets of Boolean logics, where these sets have a structure similar to the structures generally given to quantum logic. In all of these versions of quantum logic, a fundamental relation, the relation of compatibility or commutativity, plays a decisive role.