Margenau-Hill operator valued measures and joint measurability

TL;DR

This paper develops a framework using Margenau-Hill operator valued measures to analyze measurement incompatibility in quantum spin systems, providing bounds and conditions for joint measurability across different quantum dimensions.

Contribution

It introduces a fuzzy quasi measurement operator based on the Margenau-Hill rule to quantify and analyze measurement incompatibility and joint measurability in quantum spin observables.

Findings

Positivity of the MH-QMO bounds unsharpness parameters for measurement compatibility.

Degree of compatibility matches necessary and sufficient conditions for qubits.

Incompatibility increases with Hilbert space dimension.

Abstract

We employ the Margenau-Hill (MH) correspondence rule for associating classical functions with quantum operators to construct quasi-probability mass functions. Using this we obtain the fuzzy one parameter quasi measurement operator (QMO) characterizing the incompatibility of non-commuting spin observables of qubits, qutrits and 2-qubit systems. Positivity of the fuzzy MH-QMO places upper bounds on the associated unsharpness parameter. This serves as a sufficient condition for measurement incompatibility of spin observables. We assess the amount of unsharpness required for joint measurability (compatibility) of the non-commuting qubit, qutrit and 2-qubit observables. We show that the {\em degree of compatibility} of a pair of orthogonal qubit observables agrees perfectly with the necessary and sufficient conditions for joint measurability. Furthermore, we obtain analytical upper bounds on the unsharpness parameter specifying the range of joint measurability of spin components of qutrits and pairs of orthogonal spin observables of a 2-qubit system. Our results indicate that the measurement incompatibility of spin observables increases with Hilbert space dimension.