It has been seen a long ago (http://prl.aps.org/abstract/PRL/v60/i14/p1351_1) that in a pre and post-selected ensemble, the measurement of some observable on the considered system yields exotic average value (weak value) for the observable, in weak measurement limit. Since then weak values are experimentally verified despite a debate over its physical interpretation (these isues partially addressed and solved). Also weak values are used as tools in many disciplines like resolving Hardy's paradox (important to the foundations of quantum mechanics), detecting tiny effects, measuring wavefunction of single photon directly (unlike tomography) and many more. In all these cases, though measurements were weak but the most important fact was the post-selection. Recently it is employed to the parameter estimation in metrology, where it was found that in case of pre and postselection, measurements to determine the parameter of interest leads to the increased Fisher information. The inverse of the Fisher information times the success probability is the lower bound on the variance of the error in the parameter estimation. But due to decreased probabilty of success as a consequence of postselection leads to no further tightening of the lower bound despite the increased Fisher information (http://arxiv.org/abs/1306.2409, http://arxiv.org/abs/1310.5302, ). So, overall, post-selection doesn't help here to estimate the parameter of interest more precisely than that of usual quantum mechanics without post-selection. I ask the following question: Can we know a priory the situations where the additional post-selection will lead to something useful?
I think you raise a good point about the usefulness of weak-value measurements for parameter estimation, since the low probability of post-selection would seem to undermine the efficiency of weak value measurement (that is to say, why not use fewer strong measurements?)
A case where weak measurements may be useful is non-invasive (or less-invasive) diagnostics of a quantum state. My colleagues and I have recently published on using weak-value measurements for feed-forward control of a self-correcting quantum random number generator [1]. Using weak-value measurements permit us to 'diagnose' bias in the QRNG quantum state without necessarily destroying that state.
Although strong measurements could recover the same data, weak measurements preserve the (slightly perturbed) quantum state. Non-post-selected states can still be used for random number generation, while post-selected states are used for diagnostics. The strong measurement equivalent would require partitioning the ensemble of input states prior to making a diagnostic measurement.
Does this use of weak measurement meet the definition of 'useful', i.e., compared to a strong measurement alternative? I would argue yes, but that is because I prioritize preservation of the quantum state over the number of measurements needed for parameter estimation.
1. http://www.opticsinfobase.org/abstract.cfm?URI=QIM-2013-W6.37 (attached)
Thank you sir for your kind response. The point that the weak measurement doesn't destroy the state completely, is always there for the one who is using weak measurement to discuss something. But as you have already said, weak measurement does change the state (may be a little, depending on the strength of the measurement). So in your case the random numbers generated using the initial state should be different from the ones that are generated using perturbed state and therefore there must be a trade off between the measurement strength and the quality of the random numbers generated using perturbed states. It is interesting to know about your paper.
But in case of parameter estimation, the strategy that is being used focuses more on the post selection rather on weak measurement. In this situation a postselection is being done after each measurement of some estimator. So finally we don't get a state that can be reused, at least in this example. What I was seeking is, " can we get around with the decreased probability of postselection"?
Kind regards.
Thanks for clarifying that the question is about getting around the small post-selection probability. I don't have a good answer but I would mention something that might be useful. I have recently read about the balance between information gain and measurement reversibility (link below). As I understand, the more reversible the measurement, the less information is gained from the measurement
I raise this point because I wonder if the post-selection probability must not be bounded by the information gain per measurement. Consequently, the post-selection probability could be increased, but at the expense of reversibility. This might also just be a way of saying that higher order terms in the 'weak' interaction would have to be included.
I may be very, very wrong but the idea that the information gain constrains the post-selection probability makes some sense to me.
I'll be interested in your thoughts about this.
Referenced paper:
Balance between information gain and reversibility in weak measurement
Yong Wook Cheong, Seung-Woo Lee
http://arxiv.org/abs/1203.4909
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