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Applications of Mathematics

On joint distribution in quantum logics. II. Noncompatible observables

Anatolij Dvurečenskij

Published Jan 1, 1987• 1 citation
Publisher

Abstract

This paper i a continuation of the first part under the same title. The author studies a joint distribution in $\sigma$-finite measures for noncompatible observables of a quantum logic defined on some system of $\sigma$-independent Boolean sub-$\sigma$-algebras of a Boolean $\sigma$-algebra. We present some necessary and sufficient conditions fot the existence of a joint distribution. In particular, it is shown that an arbitrary system of obsevables has a joint distribution in a measure iff it may be embedded into a system of compatible observables of some quantum logic. The methods used are different from those developed for finite measures. Finally, the author deals with the connection between the existence of a joint distribution and the existence of a commutator of observables, and the quantum logic of a nonseparable Hilbert space is mentioned.

DOI
10.21136/am.1987.104275

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This paper i a continuation of the first part under the same title. The author studies a joint distribution in σ\sigmaσ-finite measures for noncompatible observables of a quantum logic defined on some system of σ\sigmaσ-independent Boolean sub-σ\sigmaσ-algebras of a Boolean σ\sigmaσ-algebra. We present some necessary and sufficient conditions fot the existence of a joint distribution. In particular, it is shown that an arbitrary system of obsevables has a joint distribution in a measure iff it may be embedded into a system of compatible observables of some quantum logic. The methods used are different from those developed for finite measures. Finally, the author deals with the connection between the existence of a joint distribution and the existence of a commutator of observables, and the quantum logic of a nonseparable Hilbert space is mentioned.
The notion of a joint distribution in σ\sigmaσ-finite measures of observables of a quantum logic defined on some system of σ\sigmaσ-independent Boolean sub-σ\sigmaσ-algebras of a Boolean σ\sigmaσ-algebra is studied. In the present first part of the paper the author studies a joint distribution of compatible observables. It is shown that it may exists, although a joint obsevable of compatible observables need not exist.
We study a conditional state on a quantum logic using Renyi's approach (or Bayesian principle). This approach helps us to define independence of events and differently from the situation in the classical theory of probability, if an event aaa is independent of an event bbb, then the event bbb can be dependent on the event aaa. We will show that we can define a sss-map (function for simultaneous measurements on a quantum logic). It can be shown that if we have the conditional state we can define the sss-map and conversely. By using the sss-map we can introduce joint distribution also for noncompatible observables on a quantum logic.
This paper deals with the topics of modeling joint distributions on a generalized probability space. An algebraic structure known as quantum logic is taken as the basic model. There is a brief summary of some earlier published findings concerning a function s-map, which is a mathematical tool suitable for constructing virtual joint probabilities of even non-compatible propositions. The paper completes conclusions published in 2020 and extends the results for three or more random variables if the marginal distributions are known. The existence of a (n+1)-variate joint distribution is shown in special cases when the quantum logic consists of at most n blocks of Boolean algebras.
We provide necessary and sufficient conditions for several observables to have a joint distribution. When applied to the bivalent observables of a quantum correlation experiment, we show that these conditions are equivalent to the Bell inequalities, and also to the existence of deterministic hidden variables. We connect the no-hidden-variables theorem of Kochen and Specker to these conditions for joint distributions. We conclude with a new theorem linking joint distributions and commuting observables, and show how violations of the Bell inequalities correspond to violations of commutativity, as in the theorem.