Exact Pointer Properties for Quantum System Projector Measurements with Application to Weak Measurements and Their Accuracy

TL;DR

This paper derives exact pointer states and measurement expressions for quantum projector measurements, including weak measurements, revealing how post-selection influences measurement outcomes and sensitivities across interaction strengths.

Contribution

It provides exact analytical expressions for pointer states and measurement results in quantum projector measurements, applicable to both strong and weak regimes, including post-selected systems.

Findings

Exact pointer states are derived for PS and PPS measurements.

Expressions for pointer distributions and mean values are valid for any interaction strength.

Measurement sensitivities for weak measurement accuracy are analyzed.

Abstract

Exact pointer states are obtained for projection operator measurements performed upon pre-selected (PS) and upon pre- and post-selected (PPS) quantum systems. These states are used to provide simple exact expressions for both the pointer spatial probability distribution profiles and the mean values of arbitrary pointer observables associated with PS and PPS projection operator measurements that are valid for any strength of the interaction which couples a measurement pointer to the quantum system. These profiles and mean values are compared in order to identify the effects of post-selection upon projector measurement pointers. As a special case, these mean value results are applied to the weak measurement regime - yielding PS and PPS mean value expressions which are valid for any operator (projector or non-projector). Measurement sensitivities which are useful for estimating weak measurement accuracies for PS and PPS systems are also obtained and discussed.