Robust weak-measurement protocol for Bohmian velocities

TL;DR

This paper introduces a robust protocol for accurately measuring Bohmian velocities using two position measurements separated by a finite time, effective under specific conditions, with differences from true velocities being minimal.

Contribution

The paper proposes a new measurement protocol for Bohmian velocities based on positive operator valued measures, demonstrating high accuracy and robustness under certain conditions.

Findings

Measured velocity differs from true velocity by less than 1% in many cases

Protocol is robust when measurement uncertainty over time interval is large

Measured velocity corresponds to the final time, not an average

Abstract

We present a protocol for measuring Bohmian - or the mathematically equivalent hydrodynamic - velocities based on an ensemble of two position measurements, defined from a Positive Operator Valued Measure, separated by a finite time interval. The protocol is very accurate and robust as long as the first measurement uncertainty divided by the finite time interval between measurements is much larger than the Bohmian velocity, and the system evolves under flat potential between measurements. The difference between the Bohmian velocity of the unperturbed state and the measured one is predicted to be much smaller than 1% in a large range of parameters. Counter-intuitively, the measured velocity is that at the final time and not a time-averaged value between measurements.