Simultaneous weak measurement of non-commuting observables: a generalized Arthurs-Kelly protocol

摘要

In contrast to a projective quantum measurement in which the system isprojected onto an eigenstate of the measured operator, in a weak measurementthe system is only weakly perturbed while only partial information on themeasured observable is obtained. A full description of such measurement shoulddescribe the measurement protocol and provide an explicit form of themeasurement operator that transform the quantum state to its post measurementform. A simultaneous measurement of non-commuting observables cannot beprojective, however the strongest possible such measurement can be defined asproviding their values at the smallest uncertainty limit. Starting with theArthurs and Kelly (AK) protocol for such measurement of position and momentum,we derive a systematic extension to a corresponding weak measurement alongthree steps: First, a plausible form of the weak measurement operator analogousto the Gaussian Kraus operator often used to model a weak measurement of asingle observable is obtained by projecting a na"ive extension (valid forcommuting observable) onto the corresponding Gabor space. Second, we show thatthe so obtained set of measurement operators satisfies the normalizationcondition for the probability to obtain given values of the position andmomentum in the weak measurement operation, namely that this set constitutes apositive operator valued measure (POVM) in the position-momentum space.Finally, we show that the so-obtained measurement operator corresponds to ageneralization of the AK measurement protocol in which the initial detectorwavefunctions is suitable broadened.