Verification of probabilities for various states

[StevieTNZ]

The experimental team takes a sample of the population to probe. I know QM can predict an expectation value for observable A. When the results are returned, there is an experimental value +/- uncertainty. For a verification, would the result need to concide with the expectation value? Would there be any uncertainty attached to the calculated expectation value?

Does the expectation value indicate how many times state a should occur, state b occur, etc, or is the probability that state a should occur (say 0.4) verified by how many quantum systems from the ensemble take on that definite state (i.e. separate from verifying the expectation value)?

Does the entanglement of the experimental systems with the 'outside world' (external from the experiment) need to be taken into account, or can we safely ignore them? I ask because if there is entanglement with everything external from the experiment, the external world is constantly changing (forces acting on the Schrödinger equation) so you won't be probing an idential wave function each time (detector entangled to external world -- the detector becomes entangled with the system it is probing as a consequence of the linearity of the equation).

 

[PatrickPowers]

The experimental team takes a sample of the population to probe. I know QM can predict an expectation value for observable A. When the results are returned, there is an experimental value +/- uncertainty. For a verification, would the result need to concide with the expectation value? Would there be any uncertainty attached to the calculated expectation value?

There is quantum uncertainty. This means that you can prepare the experiment exactly the same way every time to the best of your ability, but you still get a distribution of results.

There is something called sampling error. If you could do the experiment perfectly an infinite number of times you would get the exact distribution. With a finite number of experiments you can get as close as you can afford.

Then there is plain old experimental error. You can't do the experiment perfectly, so even if you could do it an infinite number of times you wouldn't get the exact distribution.

Does the expectation value indicate how many times state a should occur, state b occur, etc, or is the probability that state a should occur (say 0.4) verified by how many quantum systems from the ensemble take on that definite state (i.e. separate from verifying the expectation value)?

It seems to me that these are two ways of saying the same thing.

Does the entanglement of the experimental systems with the 'outside world' (external from the experiment) need to be taken into account, or can we safely ignore them? I ask because if there is entanglement with everything external from the experiment, the external world is constantly changing (forces acting on the Schrödinger equation) so you won't be probing an idential wave function each time (detector entangled to external world -- the detector becomes entangled with the system it is probing as a consequence of the linearity of the equation).

Entanglement with the outside world can't be taken into account. Everything is entangled with everything else, ever since the Big Bang. The idea is that this entanglement doesn't show any systematic trends so it can't be distinguished from sampling error.

 

[StevieTNZ]

How big a sample would the experiment need to have to start seeing verification of quantum predictions?

Is there a way to calculate roughly when convergence to the probability frequency occurs (saw the Chebyshev's inequality, but not sure if that'll do it)?