Quantum measurement paradox

Question

let's define 'a measurement device' as a system which is highly sensitive to the eigenstate of an observable. The sensitivity is quantified let's say by how irreversible and grand the small changes in the eigenstate result in the large scale, classical system's future. A wavefunction collapses when it interacts with such a device.

This seems like a simple and necessary definition, right?

But there's a problem with this interpretation. How do we calculate the 'sensitivity' of a given device, without first knowing when the wavefunction collapses? For example, in the double slit experiment, let's say that the wavefunction collapses into a sharp peak when it passes the slits. Then surely a small change in this eigenfunction will result in large changes in what happens at the screen? If the wavefunction collapses at the slits, then we can draw a midway line at the screen, and effectively use the screen as a which way device. However, since it does not collapse, we know that the screen is therefore not sensitive to the eigenstate of the wavefunction at the slits. Therefore the wavefunction should not collapse at the slits rather than at the screen.

The question is, this is clearly a case of circular logic. How do we know a priori what device will collapse the wavefunction when it passes.

Answer 1

I think what you're missing is that there is also a wavefunction collapse when a detector fails to detect something. If the wavefunction is a superposition of position states A, B, and C, and there's a detector at A and none at B or C, then there is always a collapse. It leaves the wavefunction either in whatever state the detector leaves it or else in a superposition of B and C. If the detector isn't perfect and may miss the particle, then there is still always a collapse, leaving the wavefunction either in the detected state or in a superposition of A, B, and C, where the amplitude of A is lower than before (and B and C are higher, so it's still normalized).

If there is more than one detector (say A and B are pixels on the screen, and C misses the screen entirely), you can still always think of the detectors as attempting detection one by one, succeeding with a probability given by the Born rule, and leaving the wavefunction in a changed state whether they succeed or not. You'll get the same answer regardless of the order in which you consider the measurements, at least if they're spacelike separated.

This is called interaction-free measurement. That Wikipedia article gives the impression that it's a rare thing happening only in certain experiments, but it actually happens all the time.

Answer 2

One need not to calibrate the sensitivity of the device in the experiment itself - it can be done beforehand using the appropriate eigenstates.

E.g., we can calibrate the device measuring position using particles in position eigenstates, the device measuring momentum using particles in momentum eigenstates, and then use the two devices to study something prepared in an energy eigenstate of a harmonic oscillator, where both position and momentum are uncertain.