Schrödinger's Killer App: What are the possible outcomes of the quantum pocket watches? I'm a theroretical computer scientist trying to get an intuition for quantum computing, and I'm
currently working through "Schrödinger's Killer App"[1] 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 Analogy
The pocket watches are explained in detail on p. 12 of [1]. 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.
The Question
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?
[1] 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
 A: What you've first gotta accept is that the idea of a "wave function collapsing" is only one way to interpret measurements of quantum phenomena. It is known as the copenhagen interpretation. But there are many other interpretations you may use, and it is currently not known how to determine which interpretation is optimal. It's an interesting philosophical problem called "the measurement problem." It is also related to the fact that the axioms of quantum theory are ad-hoc postulates based on experimental results, without any real underlying understanding of why. Therefore, you must be careful when reading popular treatments of quantum theory, especially with regard to the more conceptually challenging things like entangled quantum states: that is, you don't want to confuse your confusions about the philosophical issues with your confusions about the technical issues. So, just a fair warning, that if you read a different book then you might have confusions stemming from a different interpretation, but maybe not due to the physics of entanglement itself, which is pretty well understood these days (except at the cutting-edge frontier of course, but that's not what you're asking about).
So, if you measure the ion and interpret it as the "wave function collapsing" all that means, essentially, is that you're assuming there is some $\it{a}~\it{priori}$ probability distribution for the relative orientation of the ion and the act of your observer looking at it corresponds to some actualized possibility among that probability distribution.
There are other ways to interpret measurement in quantum theory, but I do not want to cause further confusion for you.
A: I think the analogy is just wrong, or so loose that it's practically useless.
Of the watches without slits, the book says

For each run of this experiment, the directions the hour hands point to will be completely random, possibly any of the 12 hours, all over the watch dial, no matter how carefully the observation is made. The only constraint is that the hands always point in opposite directions; which opposite direction they point to on any given run of the experiment is completely unpredictable.

You could construct a quantum state with that property, but it wouldn't be the EPR state that this thought experiment is supposed to be about. There's no measurement the pocket watches could be doing on an EPR state that would produce that result.
Also, these watches are essentially classical. No amount of data gathering can refute the hypothesis that the hands were simply set to a random but consistent time at the factory. If all you had were these watches, there would be no reason to believe that quantum mechanics is correct.
Since the book starts with these non-EPR, non-quantum watches, you're naturally going to think that the watches with slits in the lids are just like the previous watches, but now with slits added to the lids.
In fact, though, the watches with slits have a different state inside (an actual EPR state), and they're doing different measurements on it. Also, crucially, the measurement that they do depends on the position of the slit. When Alice and Bob choose different angles, they're doing different experiments and getting results that aren't directly comparable.
The book further confuses the issue by making the slits cover only a radius, so the possible experimental outcomes are "hand visible" and "hand invisible". You're naturally going to guess that if the hand isn't visible then it could be pointing in any other direction, since that's how the previous iteration of the watch worked. In fact, if the hand is not visible, it's necessarily pointing 180° away from the visible direction. These watches are nothing like the watches without the slits. It would have been far clearer to make the slit a diameter.
The best way to save this analogy would be to completely delete the description of the watches without slits (and also the classical synchronized watches that actually keep time, since that has nothing to do with the EPR state either), and just accept that the watches with slits behave as they behave.
