Originally posted by KazetNagorra
Could you enlighten me as to why a deterministic quantum measurement is necessarily a hidden variable theory (you don't have to hold back on the mathematics if required)? It was my impression that quantum measurements were simply not well-understood and might be an emergent phenomenon from more fundamental quantum theory (i.e. interactions between few particles rather than a macroscopic interaction).
Assuming you accept Bell's Theorem then it's not mathematical. My argument depends on Bell's assumptions about "hidden variable theories", basically that they have to be consistent and preserve local reality (in other words the momentum and position are both always well defined). The hidden variable doesn't
have to be hidden, it just has to determine the measurable variables. His inequality depends only on those two assumptions.
Essentially it is a mater of whether the outcome of a single experiment can be predicted in advance - in a hidden variable theory it can, provided you have knowledge of the hidden variable. Even if you know all the information available about a quantum system you still cannot predict the exact result of a single experiment. A theory of fundamental measurements which is deterministic has to have things like the position "existing" before the measurement is made, otherwise it wouldn't be deterministic. If the momentum or position don't exist before the measurement then it isn't a hidden variable theory, and is not deterministic. If the quantum measurement process is deterministic, then the outcome of an experiment "exists" before the measurement happens (in the same way that the point where a classical projectile hits the ground is known before the shell is fired) which means that the theory fulfills Bell's requirements for hidden variable theories.
What this means is that although the theory of quantum measurements doesn't really exist, we know that candidate theories must have the property that they do not violate Bell's inequality. Which basically means that they have to non-deterministic.
You've got a good point about quantum measurement. It's difficult to imagine an experimental apparatus capable of measuring either the position or the momentum of a particle. Spin and polarisation of light is easy as you can just use polaroid or deflect the particle in a magnetic field, and then detect the presence of the particle in a region which would be forbidden if it had the wrong spin (or polarisation). How exactly does one measure the position of a particle though? Is this the position just before, just after, or at the exact moment you bounced a photon off it? Also since the measurement involves a particle interaction, it's not clear which particle's position is being measured.