I read a number of different experiments but in reality I stopped taking it seriously when I read this:
Steinberg says that physicists have been taught that "asking where a photon is before it is detected is somehow immoral".

"Immoral?" Really?
Then we have this, for "
weak measurement"
where the initial and final states are preselected. The weak value of the observable becomes large when the postselected state approaches being orthogonal to the preselected (initial) 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.

I have two issues with this definition of weak measurement. First, this is simply the definition of how to compute the matrix representation of the operator A. Nothing has changed by calling it the result of a weak measurement. This is standard for QM. Second: If we have the complete set of orthogonal vectors (ie. the vector space is spanned by the set { phi _ i >}, then we can choose
any linear combination of final states  phi_f > that is orthogonal to  phi _i > as the initial state is orthogonal to the space of the remaining vectors. For example, if we have three vectors in Euclidean 3space, the usual i, j, and k unit vectors, any choice of linear combinations of i and j are orthogonal to k. I don't understand what they mean by "approaching."
Obviously I don't understand weak measurements, but I have two comments to make. First: How can they a priori chose the final state in the double slit experiment? It seems to me that the S
ternGerlach experiment would be a more fruitful approach since you are choosing the final state just by picking a point on a screen to measure at. Second: If you are talking about a set of mutually orthogonal base vectors (as is mentioned) then all you are doing is diagonalizing A. That means you are measuring the eigenvalues of a diagonal matrix, which means in turn that you are measuring the likelihood of the system to be in a particular state, which means in turn you are measuring the probability a certain final state will be in.
I'm not going to say that the method is bogus, but I am not going to be changing my mind as to whether this method will "measure" the wavefuntion of a particle...It will measure the
probability the particle is in a given state.
Dan