Analysis of quantum-state disturbance in a protective measurement of a spin-1/2 particle

TL;DR

This paper provides a detailed analysis of how protective measurements on a spin-1/2 particle affect the quantum state, deriving exact expressions for disturbance and exploring experimental constraints and fundamental properties.

Contribution

It introduces exact and perturbative formulas for state disturbance in protective measurement of a spin-1/2 particle, enhancing understanding and guiding experimental implementation.

Findings

Exact and perturbative expressions for state disturbance derived.

Constraints on magnetic field strengths for minimal disturbance identified.

Multiple magnetic field measurements produce similar disturbance as single superposed field measurement.

Abstract

A distinguishing feature of protective measurement is the possibility of obtaining information about expectation values while making the disturbance of the initial state arbitrarily small. Quantifying this state disturbance is of paramount importance. Here we derive exact and perturbative expressions for the state disturbance and the faithfulness of the measurement outcome in a model describing a spin-1/2 particle protectively measured by an inhomogeneous magnetic field. We determine constraints on the experimentally required field strengths from bounds imposed on the allowable state disturbance. We also demonstrate that the protective measurement may produce an incorrect result even when the particle's spin state is unaffected by the measurement, and show that successive measurements using multiple magnetic fields produce the same state disturbance as a single measurement involving a superposition of these fields. Our results supply comprehensive understanding of a paradigmatic model for protective measurement, may aid the experimental implementation of the measurement scheme, and illustrate fundamental properties of protective measurements.