Determining complementary properties using weak-measurement: uncertainty, predictability, and disturbance

TL;DR

This paper explores how weak measurements can simultaneously probe position and momentum with high resolution, challenging traditional limits by balancing uncertainty, predictability, and disturbance.

Contribution

It demonstrates that joint weak-measurements can surpass Heisenberg's limit by leveraging predictability, offering new insights into measurement disturbance and precision.

Findings

Single trials can reach Heisenberg limit resolution.

Averaging over trials can surpass the Heisenberg limit.

Weak measurement affects predictability, not just uncertainty.

Abstract

It is often said that measuring a system's position must disturb the complementary property, momentum, by some minimum amount due to the Heisenberg uncertainty principle. Using a "weak-measurement", this disturbance can be reduced. One might expect this comes at the cost of also reducing the measurement's precision. However, it was recently demonstrated that a sequence consisting of a weak position measurement followed by a regular momentum measurement can probe a quantum system at a single point, with zero width, in position-momentum space. Here, we study this "joint weak-measurement" and reconcile its compatibility with the uncertainty principle. While a single trial probes the system with a resolution that can saturate Heisenberg's limit, we show that averaging over many trials can be used to surpass this limit. The weak-measurement does not trade-away precision, but rather another type of uncertainty called "predictability" which quantifies the certainty of retrodicting the measurement's outcome.