Approximating incompatible von Neumann measurements simultaneously

TL;DR

This paper investigates the optimal approximation of incompatible orthogonal qubit measurements by jointly performing imprecise measurements, considering both observables and state transformations, and characterizes the best joint instrument.

Contribution

It introduces a comprehensive framework for optimal joint measurements of incompatible qubit observables, including state transformations, and characterizes the optimal instrument as the Luders instrument of the optimal joint observable.

Findings

Characterization of the optimal joint instrument as the Luders instrument.

Identification of the least imprecise joint measurement for orthogonal qubit measurements.

Extension of previous work by including conditional state transformations.

Abstract

We study the problem of performing orthogonal qubit measurements simultaneously. Since these measurements are incompatible, one has to accept additional imprecision. An optimal joint measurement is the one with the least possible imprecision. All earlier considerations of this problem have concerned only joint measurability of observables, while in this work we also take into account conditional state transformations (i.e., instruments). We characterize the optimal joint instrument for two orthogonal von Neumann instruments as being the Luders instrument of the optimal joint observable.