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

On joint distribution in quantum logics. I. Compatible observables

Anatolij Dvurečenskij

The notion of a joint distribution in $\sigma$ -finite measures of observables of a quantum logic defined on some system of $\sigma$ -independent Boolean sub- $\sigma$ -algebras of a Boolean $\sigma$ -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.The notion of a joint distribution in $\sigma$-finite measures of observables of a quantum logic defined on some system of $\sigma$-independent Boolean sub-$\sigma$-algebras of a Boolean $\sigma$-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.

Applications of MathematicsJan 1, 19872 citations

Margenau-Hill distribution as a Necessary and Sufficient Signature of Measurement Incompatibility

Partha Patra, A. K. Pan

Measurement incompatibility and the negativity of quasi-probability distributions both demonstrate the signature of nonclassicality. However, their relations largely remained qualitative, and no explicit operational connection has been established. We provide an operational link between measurement incompatibility and the Margenau-Hill(MH) quasi-probability distribution associated with two dichotomic observables. { We first derive the joint measurability (measurement compatibility) condition for any pair of dichotomic observables in arbitrary finite dimension $d$ .} We then rigorously prove that for any pair of unsharp dichotomic measurements in dimension $d$ , the positivity of the MH distribution is equivalent to joint measurability \emph{i.e.}, the MH distribution is positive \emph{if and only if} the measurements are jointly measurable. We further introduce the MH-like quasi-probability distribution for $n$ dichotomic unsharp observables in arbitrary finite dimension and derive a sufficient condition of joint-measurability when the observables are mutually anticommuting. Finally, we propose an interferometric setup, inspired by quantum-switch architectures, that directly reconstructs MH quasi-probability which in turn provides a test of incompatibility.Measurement incompatibility and the negativity of quasi-probability distributions both demonstrate the signature of nonclassicality. However, their relations largely remained qualitative, and no explicit operational connection has been established. We provide an operational link between measurement incompatibility and the Margenau-Hill(MH) quasi-probability distribution associated with two dichotomic observables. { We first derive the joint measurability (measurement compatibility) condition for any pair of dichotomic observables in arbitrary finite dimension $d$.} We then rigorously prove that for any pair of unsharp dichotomic measurements in dimension $d$, the positivity of the MH distribution is equivalent to joint measurability \emph{i.e.}, the MH distribution is positive \emph{if and only if} the measurements are jointly measurable. We further introduce the MH-like quasi-probability distribution for $n$ dichotomic unsharp observables in arbitrary finite dimension and derive a sufficient condition of joint-measurability when the observables are mutually anticommuting. Finally, we propose an interferometric setup, inspired by quantum-switch architectures, that directly reconstructs MH quasi-probability which in turn provides a test of incompatibility.

quant-pharXiv:2609.37144Sep 30, 20260 citations

Compatibility and Marginality

Oľga Nánásiová, Andrei Yu. Khrennikov

International Journal of Theoretical PhysicsApr 25, 200711 citations

Conflicts and the Basic Incompatibility

Max Dresden

Springer eBooksJan 1, 19870 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.

The notion of a joint distribution in

\sigma

-finite measures of observables of a quantum logic defined on some system of

\sigma

-independent Boolean sub-

\sigma

-algebras of a Boolean

\sigma

-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.

d

.} We then rigorously prove that for any pair of unsharp dichotomic measurements in dimension

d

, the positivity of the MH distribution is equivalent to joint measurability \emph{i.e.}, the MH distribution is positive \emph{if and only if} the measurements are jointly measurable. We further introduce the MH-like quasi-probability distribution for

n

dichotomic unsharp observables in arbitrary finite dimension and derive a sufficient condition of joint-measurability when the observables are mutually anticommuting. Finally, we propose an interferometric setup, inspired by quantum-switch architectures, that directly reconstructs MH quasi-probability which in turn provides a test of incompatibility.