Conditions necessary for experimental conflict with Bell's Theorem In Aspect 2002 - Bell's Theorem, the Naive view of an Experimentalist, he suggests conditions necessary for coming up with situations/experiments that can conflict with Bell's inequality. I'm interested in explanations for the two conditions he line up:

Moreover, in situations usually described by quantum mechanics, it does not often happen that there is a conflict with Bell’s inequalities. More precisely, for situations in which one looks for correlations
between two separated subsystems (that may have interacted in the
past), we can point out two conditions necessary to have the
possibility of a conflict with Bell’s inequalities:
• 1) The two separated subsystems must be in an entangled state,
non-factorizable
• 2) For each subsystem, it must be possible to choose the measured
quantity among at least two non-commuting observables (such as
polarization measurements along directions a and a’, neither parallel
nor perpendicular)

I know what a Bell-state is and I know what non-commuting observables are which are usually used in the Bell-type experiments (polarizer angles that are not orthogonal or parallell) but I'm trying to go deeper towards the "root cause" as it were. For example, for 2) above, for a CHSH-type experiment you actually measure 4 correlations between two non-commuting polarizer angles for detector A and two angles for detector B and sum them in the inequality statement - why is the equality satisfied if only one or two of the 4 pairs are summed? And why don't you get any conflict if the angles are not chosen in a specific range? I know the equations and I've seen the plots, but I'm interested in the causes, if it's possible to drill down more..

 A: With each measurement, the state of the observables is destroyed. What does that mean? If I measure the polarization angle, I change it.
Let's assume that the photon is polarized vertically. With a polarization filter at 45°, 50% of the time the photon will be absorbed and 50% of the time its plane of polarization will be rotated to the 45° and it will appear on the other side of the polarizer (1). If the filter is brought more into vertical alignment, the ratio of transmitted to blocked photons will improve.
Now it is so that we are empirically already convinced to have generated entangled photons whose electric fields are entangled by 90° to each other. What we (not yet?) have under control is the concrete arrangement of, for example, 0° and 90°. We get only x° and x°+90° (forgive me the ° after the x).
This means that our knowledge about the entanglement of the particles arises statistically. Several measuring processes are necessary to decide whether the photons come from an entangling source or not. Or, whether the communication was overheard.
(1) If someone has not made the experience yet: Two filters arranged one behind the other with 90° twist to each other, block all light. A filter in between under optimally 45° lets light through again.
That's not an answer, but it's too long for a comment.
