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

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 'weakmeasurement', 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.

On the validity of weak measurement applied for precision measurement

“…This limiting value is expected to coincide with the theoretical expectation value ζ|X|ζ when a perfectly accurate theoretical model is available. We then classify the difference (17) in two components of errors according to the inequality,…”

Uncertainty Relations: Curiosities and Inconsistencies

“…In general, a common feature of all these cases is that the uncertainty relation takes the form considered in this paper, that is the form given by inequalities (1), (2) and (4). A discussion of different aspects of these interpretations as well as attempts to improve uncertainty relations are still continued and can be found in many papers (see, e.g., [29][30][31][32][33]).…”

Theory and experiment for resource-efficient joint weak-measurement

“…As shown in Ref. [12], as one decreases the disturbance caused by the measurement process, one also decreases the 'predictability' of the measurement. Weak measurement has a broad range of applications from amplifying tiny signals [13][14][15] to fundamental studies on the meaning of a quantum state [16,17].…”

“…For instance, for a photon there are just four DOF: polarization, and a threedimensional wavevector (which, in turn, incorporates frequency-time and transverse positionmomentum). Due to this limitation, joint weakmeasurement experiments have never progressed beyond the product of two observables [7,12]. To overcome this constraint, the present work theoretically introduces and experimentally demonstrates a technique to perform a joint weakmeasurement of multiple observables using a single DOF as the read-out system.…”