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

Maicol A. Ochoa*, Wolfgang Belzig, Abraham Nitzan

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


In contrast to a projective quantum measurement, in a weak measurement the system is only weakly perturbed while only partial information on the measured observable is obtained. A simultaneous measurement of non-commuting observables cannot be projective, however the strongest possible such measurement can be defined as providing their values at the smallest uncertainty limit. Starting with the Arthurs and Kelly (AK) protocol for such measurement of position and momentum, we derive a systematic extension to a corresponding weak measurement along three steps: First, a plausible form of the weak measurement operator analogous to the Gaussian Kraus operator, often used to model a weak measurement of a single observable, is obtained by projecting a naïve extension (valid for commuting observable) onto the corresponding Gabor space. Second, we show that the so obtained set of measurement operators satisfies the normalization condition for the probability to obtain given values of the position and momentum in the weak measurement operation, namely that this set constitutes a positive operator valued measure (POVM) in the position-momentum space. Finally, we show that the so-obtained measurement operator corresponds to a generalization of the AK measurement protocol in which the initial detector wavefunctions is suitable broadened.

Original languageEnglish
Article number15781
JournalScientific Reports
Issue number1
StatePublished - 1 Dec 2018


FundersFunder number
Israel-US Binational Science Foundation
National Science FoundationSFB 767
Eden Hall Foundation
Lion Foundation
Deutsche ForschungsgemeinschaftTH 820/11-1


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