Weak measurement
modified slightly by me in italic font

Weak measurements are a type of quantum measurement, where the measured system is very weakly coupled to the measuring device. After the measurement the measuring device pointer is shifted by what is called the "weak value", so that a pointer initially pointing at zero before the measurement would point at the weak value after the measurement. The system is not disturbed by the measurement. Although this may seem to contradict some basic aspects of quantum theory, the formalism lies within the boundaries of the theory and does not contradict any fundamental concept, in particular not Heisenberg's uncertainty principle.

The idea of weak measurements and weak values, first developed by Yakir Aharonov, David Albert and LevVaidman, published in 1988, [1] is especially useful for gaining information about pre- and post-selected systems described by the two-state vector formalism.[2] This was the original reason that Aharonov et al developed weak measurement. Since a "strong" perturbative measurement can both upset the outcome of the post-selection and tamper with all subsequent measurement, weak nonperturbative measurements may be used to learn about such systems during their evolution.

If  (history| and |destiny)  are the pre- and post-selected quantum mechanical (retarded past to present) history and (advanced back from the future) destiny states, the weak value of the observable Â is defined as

Aw = (history|A|destiny)/(history|destiny)

The weak value of the observable becomes large when the post-selected state approaches being orthogonal to the pre-selected state, . In this way, by properly choosing the two states, the weak value of the operator can be made arbitrarily large, and otherwise small effects can be amplified.[3]

Note that the theory of weak measurement allows Hardy's paradox to be explained. In Hardy's paradox a positron and an electron go down both arms of each of their interferometers. If they meet in the overlapping arms, they should annihilate each other. But, strangely, they are still registered as arriving at the detectors.[4]

Related to this, the research group of Aephraim Steinberg at the University of Toronto confirmed Hardy's paradox experimentally using joint weak measurement’ of the locations of entangled pairs of photons.[4][5] Independently, a team of physicists from Japan reported in December, 2008, and published in March, 2009, that they were able to use joint weak measurement to observe a photonic version of Hardy's paradox. In this version, two photons were used instead of a positron and an electron and relied not upon non-annihilation but on polarization degrees of freedom values measured.[6]

Building on weak measurements, Howard M. Wiseman proposed a weak value measurement of the velocity of a quantum particle at a precise position, which he termed its "naïvely observable velocity". In 2010, a first experimental observation of trajectories of a photon in a double-slit interferometer was reported, which displayed the qualitative features predicted in 2001 by Partha Ghose[7]for photons in the de Broglie-Bohminterpretation.[8][9]

In 2011, weak measurements of many photons prepared in the same pure state, followed by strong measurements of a complementary variable, were used to reconstruct the state in which the photons were prepared.[10]

Discover Magazine article: "Back From the Future" A series of quantum experiments shows that measurements performed in the future can influence the present. http://discovermagazine.com/2010/apr/01-back-from-the-future
Stephen Parrott questions the meaning and usefulness of weak measurements, as described above.[6]
Quantum physics first: Researchers observe single photons in two-slit interferometer experiment:http://www.physorg.com/news/2011-06-quantum-physics-photons-two-slit-interferometer.html
Adrian Cho: Furtive Approach Rolls Back the Limits of Quantum Uncertainty, Science, 5 August 2011, vol. 333, no. 6043, pp. 690-693, doi:10.1126/science.333.6043.690
References
^ Y. AharonovD.Z. Albert, L. Vaidman, "How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100," Physical Review Letters, 1988. [1]
^ Y. Aharonov and L. Vaidman in Time in Quantum Mechanics, J.GMuga et al. eds., (Springer) 369-412 (2002) quant-ph/0105101
^ O. Hosten and P. Kwiat Observation of the spin Hall effect of light via weak measurements Science 319 787 (2008) [2]
^ a b J. S. Lundeen, A. M. Steinberg, "Experimental Joint Weak Measurement on a Photon Pair as a Probe of Hardy’s Paradox", Physical Review Letters 102, 020404 (2009) [3]