Margenau–Hill operator valued measures and joint measurability

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 noncommuting 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 noncommuting qubit, qutrit and 2-qubit observables. We show that the 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.