Many experiments with entangled photons are sending them through different glass fiber cables (e.g. in opposite directions for spatial separation). The photons will inevitably be reflected many times in the cable before reaching the end of the fiber, correct? Why is this interaction with the optical barrier layer not a measurment, which to my understanding would destroy the entanglement?
The interaction is not a measurement because the probability that it will produce a measurable change in the momentum of the reflecting object is extremely small. The reflector is in a mixed state in which its momentum has a range of values that is large compared to the momentum of the photon. So the shift in the mirror's momentum as a result of the reflection will be so small that it will not be detectable with very high probability. I don't know the exact numbers but let's say it's -1,000,000 to +1,000,000 in units of the momentum the photon will impart upon reflection. The photon is incident on the mirror and changes its momentum by +1 so that the range is now -999,999 to +1,000,001. So the probability is large that if you measure its state you won't detect any difference.