The weakening measurement

Unlike the special theory of relativity that the superluminal-neutrinos fiasco sought to defy, Heisenberg’s uncertainty principle presents very few, and equally iffy, measurement techniques to stand verified. While both Einstein’s and Heisenberg’s foundations are close to fundamental truths, the uncertainty principle has more guided than dictated applications that involved its consequences. Essentially, a defiance of Heisenberg is one for the statisticians.

And I’m pessimistic. Let’s face it, who wouldn’t be?

Anyway, the parameters involved in the experiment were:

1. The particles being measured
2. Weak measurement
3. The apparatus

The experimenters claim that a value of the photon’s original polarization, X, was obtained upon a weak measurement. Then, a “stronger” measurement was made, yielding a value A. However, according to Heisenberg’s principle, the observation should have changed the polarization from A to some fixed value A’.

Now, the conclusions they drew:

1. Obtaining X did not change A: X = A
2. A’ – A < Limits set by Heisenberg

The terms of the weak measurement are understood with the following formula in mind:

(The bra-ket, or Dirac, notation signifies the dot-product between two vectors or vector-states.)

Here, φ(1,2) denote the pre- and post-selected states, A-hat the observable system, and Aw the value of the weak-measurement. Thus, when the pre-selected state tends toward becoming orthogonal to the post-selected state, the value of the weak measurement increases, becoming large, or “strong”, enough to affect the being-measured value of A-hat.

In our case: Aw = A – X; φ(1) = A; φ(2) = A’.

As listed above, the sources of error are:

1. φ(1,2)
2. X

To prove that Heisenberg was miserly all along, Aw would have been increased until φ(1) • φ(2) equaled 0 (through multiple runs of the same experiment), and then φ(2) – φ(1), or A’ – A, measured and compared to the different corresponding values of X. After determining the strength of the weak measurement thus, A’ – X can be determined.

I am skeptical because X signifies the extent of coupling between the measuring device and the system being measured, and its standard deviation, in the case of this experiment, is dependent on the standard deviation of A’ – A, which is in turn dependent on X.