Quantum Measurements

Related to: The Quantum Physics Sequence, particular Decoherence is Pointless.

If you really understand quantum mechanics (or the linked post on decoherence) you shouldn’t get anything out of this post, but understanding this really cleared things up for me, so hopefully it will clear things up for someone else too.

Here at lesswrong we are probably all good Solomonoff inductors and so tend to reject collapse. We believe that a measurement destroys interference because it entangles some degrees of freedom from a quantum system with its environment. Of course, this process doesn’t occur sharply and discontinuously. It happens gradually, as the degree of entanglement is increased. But what exactly does “gradually” mean here? Lets focus on a particular example.

Suppose I run the classic two-slit experiment, but I measure which slit the electron goes through. Of course, I don’t observe an interference pattern. What happens if we “measure it less”? Lets go to the extreme: I change the polarization of a single photon depending on which slit the electron went through (its either polarized vertically or horizontally), and I send that photon off to space (where its polarization will not be coupled to any other degrees of freedom). Do I now see just a little bit of an interference pattern?

A naive guess is that strength of the interference pattern drops off exponentially with the number of degrees of freedom entangled with the measurement of which slit the electron went through. In a certain sense, this is completely correct. But if I were to perform the experiment exactly as I described—in particular, if I were to polarize the photon perfectly horizontally in the one case and perfectly vertically in the other case—then I would observe no interference at all. (This may or may not be possible, depending on the way nature chose to implement electrons, photons, and slits. Whether or not you can measure exactly is really not philosophically important to quantum mechanics, and I feel completely confident saying that science doesn’t yet know the answer. So I guess I’m not yet decided on whether decoherence really happens “gradually.” )

The important thing is that two paths leading to the same state only interfere if they lead to exactly the same state. The two ways for the electron to get to the center of the screen interfere by default because nature has no record of how the electron got there. If you measure at all, even with one degree of freedom, then the two ways for the electron to get to the same place on the screen don’t lead to the exactly same state and so interference doesn’t occur.

The exponential dependence on the number of degrees of freedom comes from the error in our measurement devices. If I prepare one photon polarized either horizontally or vertically, and I do it very precisely, then I am very unlikely to mistake one case for the other and I will therefore see very little interference. If I do it somewhat less precisely, then the probability of a measurement error increases and so does the strength of the interference pattern. However, if I create 1000 photons, each polarized approximately correctly, then by taking a majority vote I can almost certainly correctly learn which slit the electron went through, and the interference pattern disappears again. The probability of error drops off exponentially, and so does the interference.

Another issue (which is related in my head, probably just because I finally understood it around the same time) is the possibility of a “quantum eraser” which destroys the polarized photon as it heads into space. If I destroy the photon (or somehow erase the information about its polarization) then it seems like I should see the interference pattern—now the two different paths for the electron led to exactly the same state again. But if I destroy the photon after checking for the interference pattern, how can this be possible?

The answer to this apparent paradox is that erasing the data in the photon is impossible if you have already checked for an interference pattern, by the reversibility of quantum mechanics. In order to erase the data in the photon, you need to measure which slit the electron went through a second time in a way that precisely cancels out your first measurement; there is no way around this. This conveniently prevents any sort of faster than light communication. In order to restore the interference pattern, you need to bring the photon back into physical proximity with the electron.