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Simultaneous weak measurement of non-commuting observables: a generalized Arthurs-Kelly protocol
Abstract
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.
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“In postselected weak measurement technique, the weak value of nonlocal joint operators can not be obtained exactly and the efficieny is too low for entangled state case [32].”
From Direct measurement methods of density matrix of an entangled quantum state (2020) · cites this paper as [32]
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“If we assume that the polarizers represented by the projection operator π ij and π DD causing displacement along x and y directions, respectively, we can get the weak average of π DD π ij by using the method introduced in Refs.[11,32] as”
From Direct measurement methods of density matrix of an entangled quantum state (2020) · cites this paper as [32]
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(1983) · Lecture Notes in Physics · doi:10.1007/3-540-12732-1 · Cited by 1,565
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Cited by
Sorted by citations
Direct measurement methods of density matrix of an entangled quantum state
Turek, Y. (2020) · Journal of Physics Communications · doi:10.1088/2399-6528/ab9938 · Cited by 4
Metadata
- Authors
- Ochoa, M. A., Belzig, W., & Nitzan, A.
- Venue
- Scientific Reports
- Published
- 2018-10-25
- Volume
- 8
- Issue
- 1
- ISSN
- 2045-2322
- PMC
- PMC6202392
- Cited by
- 20
- References
- 1
- Language
- en
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