How much does the collapse of the wave function reveal about the state of the quantum prior to collapse? The best way I can pose this question is through an example: suppose a photon passes through a beamsplitter, putting the photon into a superposition of the two paths (reflected or passed through), and along the two paths, equidistant from the beamsplitter, there are detectors. Now, when the photon in superposition reaches the detector on either path, it will hit one and not the other.
My question is whether or not one can conclude that the photon must have been traveling on the path which correlates to the activated detector throughout the entirety of the travel. My intuition and my idea (not expert) of how quantum objects behave are at odds. On one hand, my intuition believes that if an object is detected at the end of a given path, it must have traveled in that path in its entirety. On the other hand, I have been led to believe that the exact position of a quantum object in superposition cannot be described as a singular location, path, etc. Does the after-the-fact detection of a quantum object along a specific path prove that the quantum was traveling that path the whole time? If so, does it also prove that the quantum traveled that path exclusively?
 A: The classic experiment for discussing this issue is the Mach-Zehnder interferometer. A source of photons is passed through a beamsplitter, bounced off two mirrors, and combined again using another beamsplitter device. There are two paths photons might emerge on, and detectors are set up on both. If the path lengths are adjusted to match to an accuracy of the wavelength of the light, interference causes all the photons to emerge on one path and none on the other. Even if we reduce the light intensity so that only one photon is in the instrument at a time (so we need a photomultiplier to detect it).
The photons have to be travelling down both paths simultaneously, or the two path lengths couldn't affect the output. We can see this more clearly if we block one of the paths. Now half the photons hit the obstruction, and the other half, travelling down the other arm, see no interference at the beamsplitter and so are split both ways. The detector that previously was seeing none of the photons is now seeing lots.
Every photon this detector sees must have come down the unobstructed path. But the obstruction on the path it didn't travel down is somehow having an effect, causing these photons to be detected where there were previously none. Something must be travelling down both paths, to know what's there, even though only one photon is detected at a time, at only one of the detectors.
We can detect the obstruction with photons that could never have gone anywhere near it. The double-slit experiment has the same implication, although is a bit more subtle. In this case, the obstructed path can be miles away from the path of the photons we see, so it's a bit more obvious that something unintuitive is going on.

A: There is a famous experiment similar to yours that makes it easier to draw conclusions: the double-slit experiment.
In the double-slit experiment, one is similarly tempted to conclude that the particle is going through a single slit at a time. Nevertheless, we get interference effects, which means we must consider that the particle has traveled through both slits and had its state collapse only at the moment of detection.
