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Manfred comments on Open thread, 24-30 March 2014 - Less Wrong Discussion
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Here's a cute/vexing decision theory problem I haven't seen discussed before:
Suppose you're performing an interference experiment with a twist: Another person, Bob, is inside the apparatus and cannot interact with the outside world. Bob observes which path the particle takes after the first mirror, but then you apply a super-duper quantum erasure to Bob so that they remember observing the path of the particle, but they don't remember which path it took. Thus, at least from your perspective, the superposed versions of Bob interfere, and the particle always hits detector 2. (I can't find the reference for super-duper quantum memory erasure, probably because it's behind a paywall. Perhaps (Deutsch 1996) or (Lockwood 1989).)
Suppose that after Bob makes their observation, but before you observe Bob, you offer to play a game with Bob: If the particle hits detector 2, you give them $1; but if it hits detector 1, they give you $2. Before the experiment ran, this would have seemed to Bob like a guaranteed $1. But during the experiment, it seems to Bob that the game has expected value -$.50. What should Bob do?
If it seems unfair to wipe Bob's memory, there's an equivalent puzzle in which Bob doesn't learn anything about the particle's state, but the particle nevertheless becomes entangled with Bob's body. In that case, the super-duper quantum erasure doesn't change Bob's epistemic state.
My grasp of quantum physics is rudimentary; please let me know if I'm completely wrong.
Bob cannot become entangled with the outside world while in the middle of a quantum erasure experiment, or else it doesn't work. So he doesn't really get to do anything :P
If Bob knows that the particle becomes entangled with him, then he still makes the same predictions.
Ok, that's surprising. Here's why I thought otherwise: From Bob's perspective, a particle is prepared in a superposition of states B and C. Then Bob observes or becomes entangled with the particle, thus collapsing its state. Then the super-duper quantum erasure is performed, which preserves the state of the particle. Then the particle strikes the second half-silvered mirror. A collapse interpretation tells Bob to expect two outcomes with equal probability. Is this, then, an experiment where a collapse interpretation and a many-worlds interpretation give different predictions?
The collapse interpretation predicts that you can't do the super-duper quantum erasure. Once the collapse has occurred the wavefunction can't uncollapse.
Basically, there are a variety of collapse interpretations depending on where you make the collapse happen. Every time we've tested these hypotheses (e.g. by this sort of experiment), we haven't been able to see an early collapse.
At this point, all actual physicists I know just postpone the collapse whenever necessary to get the right answer.
Hm, so that means that quantum physics predicts that our observations depend on the presence of parallel worlds in the universal wavefunction, which in theory might interfere with our experiments at any time, right?
Calling them parallel worlds is as always dangerous (you can't go all buckaroo bonzai on them), but basically yes.
He can, in theory, make bets. Just so long as the bet he makes doesn't depend on which way he saw the particle go.
Hm, good point. We could set aside a few instants for him to send a few photons that wouldn't depend on the state of the particle. From a practical standpoint that's pretty impossible, but forget practicality.
So, sure; Bob should accept the bet. Although if he makes his answer to the bet depend on the state of the particle at all, then he shouldn't accept the bet :P There might be some interesting applications of this, say where the option "don't accept" has some positive payoff if the particle changes directions. Bob can precommit to send out an entangled qubit to get some chance at that reward.