I am looking to confirm or clarify my understanding of a measurement on an polarization entangled pair of photons. I am considering an entangled pair in the state |ϕ+⟩ =1√2(|HH⟩+|VV⟩) (positively correlated) in which one is sent to Alice and the other to Bob.
If both Alice and Bob measure their photons with a polarization beam splitter (PBS) in the HV basis, they will both get the same result. For example, if Alice measures her photon in the PBS and it randomly collapses to be H, then Bob will always measure his photon to be also H.
Likewise, if their PBSs are in the diagonal basis, if Alice measures her photon to be |+45>, then Bob will always get |+45>. If Alice measures at any arbitrary basis (say, at 15∘ / 105∘, as a random example), and her photon collapses to 15∘, then Bobs photon will be at 15∘. It is as though Bobs photon knows how Alice measured her photon and the result and orientates itsself. This is my understanding and I am just looking to clarify it in simple terms.
Taking this a little further, then, if Alice and Bob agree ahead of time a random sequence of basis measurements (even if it is just all HV) and agree how to interpret the results (eg: H=0; V=1), they will (after both measurements) instantly share the same random sequence of 1's and 0's. Again, looking to confirm/clarify my understanding of this.