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This is prompted by the strong claims made in Science 332, 1170 (2011) to have observed trajectories of photons, "something all of our textbooks and professors had always told us was impossible". I'm suspicious of this claim because they work entirely within quantum theory, and it seems to me that in making weak measurements at two points on what they say is a trajectory, they do nothing else than make unsharp measurements of incompatible observables, a concept that was first introduced, to my knowledge, by Busch and Lahti in Phys. Rev. D 29, 1634–1646 (1984), and that is now much used in quantum information. Certainly observables that are at time-like separation are in general incompatible, so an unsharp measurement approach would seem applicable.

A brief account of the Science paper mentioned above can be found here, amongst other places, and a few quotes can be found here, but, this being Science, I couldn't find a preprint. I was first alerted to this paper by a page at the BBC. The claim highlighted by ZapperZ, "Single-particle trajectories measured in this fashion reproduce those predicted by the Bohm–de Broglie interpretation of quantum mechanics (8), although the reconstruction is in no way dependent on a choice of interpretation", seems to me particularly egregious, insofar as both their choice of what experiment to do and of what to do with their raw data seem partly driven by the form they want to present their results in. It seems entirely possible to present the data in terms of unsharp measurements and POVMs, which would give relatively less support to the de Broglie-Bohm interpretation.

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I don't have a firm enough grasp of these concepts to give you a proper answer, but the weak measurements here are those of Aharonov, Albert and Vaidman. They are not POVMs. –  John Schanck Jun 3 '11 at 15:35
    
@John Schanck Should have mentioned that I read that this morning. Agreed, their account of a weak measurement doesn't look like a POVM because they describe an experiment in terms of Hamiltonian interactions between an observed system and a measurement apparatus, but if one traces out the measurement apparatus from their model one will be left with a POVM model for the experiment. This suggests a rewording of my question that better reflects my intention in asking (without which your Answer seems to me to be correct), "Is a weak measurement equivalent to an unsharp measurement or POVM?" –  Peter Morgan Jun 3 '11 at 16:03

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You are right, a weak measurement is but a special POVM (which are the most general measurement). However, Aharonov and coworkers considered more specifically a weak measurement followed by a post-selection. It is the combination of the weak interaction, the coherence of the probe ('weak' means that the coupling constant is smaller than the coherence scale of the probe), and the post-selection that allows to see interference effects in the readout of the probe. These effects can manifest, e.g., in the large average output of the probe. Since the post-selection is another measurement (usually, but not necessarily, strong), we could say that the procedure of AAV is to consider two sequential POVMs and then condition the probability of the first one upon the outcome of the second.

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