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.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.

Applications of MathematicsJan 1, 19871 citation

A Formal Theory for Finite-Dimensional Possibilistic Quantum Mechanics

Olivier Brunet

In this work, we present a logical formalism for reasoning about quantum systems in finite dimension. Contrary to the usual approach in quantum logic, our formalism is based classical first-order logic, which allows us to use the tools of model theory in our study. In particular, we show that our formal theory is complete, meaning that it entirely determines the behaviour of quantum systems. Moreover, we provide a characterization of the models of our formal theory, thus providing new insights in the study of hidden variable models of quantum theory.In this work, we present a logical formalism for reasoning about quantum systems in finite dimension. Contrary to the usual approach in quantum logic, our formalism is based classical first-order logic, which allows us to use the tools of model theory in our study. In particular, we show that our formal theory is complete, meaning that it entirely determines the behaviour of quantum systems. Moreover, we provide a characterization of the models of our formal theory, thus providing new insights in the study of hidden variable models of quantum theory.

quant-pharXiv:2602.16368Feb 19, 20260 citations

Logic of infinite quantum systems

Daniele Mundici

International Journal of Theoretical PhysicsOct 1, 199330 citations

Theory of Quantum Logic

Anatolij Dvurečenskij

Springer eBooksJan 1, 19930 citations

Quantum Logic for Quantum Computers

Mladen Pavičić

International Journal of Theoretical PhysicsMar 1, 20000 citations

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.