Uncertainty limits for quantum metrology obtained from the statistics of weak measurements

TL;DR

This paper derives a fundamental uncertainty limit in quantum metrology based on the statistics of weak measurements, linking weak values to phase sensitivity and measurement limitations.

Contribution

It introduces a new uncertainty relation for quantum metrology that incorporates weak measurement statistics, providing a time-symmetric perspective on measurement sensitivity limits.

Findings

Phase sensitivity is given by the variance of the imaginary parts of weak values.

Measurement limitations can be included by subtracting the variance of real parts of weak values.

The relation offers a time-symmetric formulation of quantum measurement uncertainty.

Abstract

Quantum metrology uses small changes in the output probabilities of a quantum measurement to estimate the magnitude of a weak interaction with the system. The sensitivity of this procedure depends on the relation between the input state, the measurement results, and the generator observable describing the effect of the weak interaction on the system. This is similar to the situation in weak measurements, where the weak value of an observable exhibits a symmetric dependence on initial and final conditions. In this paper, it is shown that the phase sensitivity of a quantum measurement is in fact given by the variance of the imaginary parts of the weak values of the generator over the different measurement outcomes. It is then possible to include the limitations of a specific quantum measurement in the uncertainty bound for phase estimates by subtracting the variance of the real parts of the weak values from the initial generator uncertainty. This uncertainty relation can be interpreted as the time-symmetric formulation of the uncertainty limit of quantum metrology, where the real parts of the weak values represent the information about the generator observable in the final measurement result.