I'm a theroretical computer scientist trying to get an intuition for quantum computing, and I'm currently working through "Schrödinger's Killer App" by the late Jonathan P. Dowling.
Dowling uses the analogy of quantum-entangled pocket watches carried by Alice and Bob to introduce the reader to the concepts of nonlocality, unreality, uncertainty and how Bell's test proves that quantum theory wins over hidden variable theory. There are some things in this analogy that I do not get, and I'm getting to a point in the book where I suspect I need to understand these points to further progress.
In the next section I quickly outline how these watches work for people unfamiliar with the book. Otherwise, you can skip to "The Question".
The pocket watches are explained in detail on p. 12 of . I'll summarize, without a guarantee that my understanding is correct: Both watches only have a single clock hand and include a calcium ion. The watches have a covering lid. When you open the lid, some apparatus measures the north pole of the ion's magnetic field and moves the clock hand to indicate the direction of the north pole. The ions of both watches have been entagled "in the right way", s.t. their magnetic poles must point in opposite directions when measured - i.e., if Alice sees 3:00, Bob sees 9:00.
In a second step, the watches are augmented with a "viewing slit", a cover on top of the clock's face that allows only to look at a single hour of the clock. Alice and Bob must each first turn the cover into a position and then open the cover, which triggers the measure-and-turn-hand mechanism. They can then only spot whether the hand points to the selected hour or not. This augmentation is described on p. 19.
I think my problem is that I have the wrong intuition of what happens to "the wave function" when a measurement is made.
On p.21f, Dowling lists the possible outcomes of the quantum-clock-with-slits-experiment of Alice and Bob. I quote (highlight by me):
The primary point is that if Alice sees “yes” in her slit, and by chance Bob has chosen the 180° slit, Bob must also see “yes”—as in test number three where Alice sees 8:00 and Bob sees 2:00. If the hands are there, visible in both slits by random chance, then they must be anti-correlated—that is, they must point in opposite directions as before.
The secondary point in Charlie’s protocol is that there are now many other things that can happen. For example, the hand may not be there at all. Or even if it is, Alice and Bob may not have chosen the slits that are 180° opposite to each other, because they each make this choice at random. Hence, Alice may see a hand and Bob nothing, or both Alice and Bob may see nothing. It is even possible that by random chance, Alice and Bob will both choose slits that are not 180° apart and both see the hour hand, but these events will happen at ran- dom with no apparent anti-correlation.
The highlighted part is what confuses me. Say Alice chose 12:00 and Bob chose 3:00. Alice opens her cover and sees the hand. In my understanding, this measures her ion's pole, thus "collapses the wave function", i.e., makes superposition of all possible states disappear and establishes a single state as the reality. However, since the ions are entangled, this also collapses the wave function for Bob's ion. Since Alice sees the ion's north pole at 12:00, Bob's ion's north pole must collapse to 6:00, right? How can it happen that Bob sees the north pole at some position that is not 180° off from Alice's position?
 There's a free copy of that book hosted at Dowling's old university page, so I assume it's okay to link to this copy: http://www.phys.lsu.edu/~jdowling/publications/SchroedingersKillerApp.pdf