Approximate joint measurement of qubit observables through an Arthur-Kelly type model

TL;DR

This paper explores the joint measurement of multiple unsharp qubit observables using an Arthur-Kelly type model, analyzing the effects of detector states, measurement unsharpness, and system-detector entanglement, with implications for measurement uncertainty and symmetry considerations.

Contribution

It introduces a detailed analysis of joint measurements of unsharp qubit observables within an Arthur-Kelly framework, highlighting the role of initial detector states and symmetries in measurement properties.

Findings

Connection between POVM elements and Hamiltonian symmetries clarified.

Derived a necessary condition for approximate joint measurement of three observables.

Demonstrated the influence of initial detector states on measurement unsharpness and system post-measurement states.

Abstract

We consider joint measurement of two and three unsharp qubit observables through an Arthur-Kelly type joint measurement model for qubits. We investigate the effect of initial state of the detectors on the unsharpness of the measurement as well as the post-measurement state of the system. Particular emphasis is given on a physical understanding of the POVM to PVM transition in the model and entanglement between system and detectors.Two approaches for characterizing the unsharpness of the measurement and the resulting measurement uncertainty relations are considered.The corresponding measures of unsharpness are connected for the case where both the measurements are equally unsharp. The connection between the POVM elements and symmetries of the underlying Hamiltonian of the measurement interaction is made explicit and used to perform joint measurement in arbitrary directions. Finally in the case of three observables we derive a necessary condition for the approximate joint measurement and use it show the relative freedom available when the observables are non-orthogonal.