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Put the classical electron-emitting double-slit apparatus in a sealed box. At each slit there's a counter to check whether an electron has passed it or not, so had the apparatus been left in open, we would have observed dots pile after the two slits on the screen with no interference pattern. But now the apparatus is running in the sealed box, and the box is so designed that 1) an outside observer can't know anything inside the box unless he opens it. 2) the box destroys the counters just before you open it, leaving no information of their past recordings. My question is, will you see interference pattern after you open the box? Why or why not?

To quote Feynman:"You do add the amplitudes for the different indistinguishable alternatives inside the experiment... At the end of the process you may say that you 'don't want to look at the photon.' That's your bussiness... Nature does not know what you are looking at, and she behaves the way she's going to behave whether you bother to take down the data or not" Does this comment apply here? But which slit the electrons have passed does seem indistinguishable to an outside observer during or after the experiment.

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Yes, Feynman's comments apply to this situation – it's really the very situation Feynman was referring to. The counters destroy the interference pattern whether or not you bother to look. In fact, the claim that "you cannot look" is only valid in practice, not in principle. In principle, the state of the counters before their destruction inevitably leaves some traces in the environment (although traces that are hard to observe in practice) because the information is preserved.

By inserting the counters, we're correlating the state of the electron with a state of the counter and this entanglement with some degrees of freedom in the environment (in the counter) is the immediate reason why the interference is no longer there.

When we say that the right interpretation of the wave function is subjective, it doesn't change anything about the fact that counters behind slits disrupt the interference pattern (and similar consequences). Instead, we may say that without an observer, the whole sealed box evolves into a Schrödinger-cat-like superposition of macroscopically distinct states.

But when we look at the superposition in your case, it contains contributions from the electron in one slit with one state of the counter; and an electron in the right slit with another state of the counter. Each of these possibilities has a nonzero probability amplitude – and nonzero probability – and once we open the (partly destroyed) sealed box, we measure which of the two outcomes has gotten realized. But we may still show that we will measure the strength of the interference pattern to be zero if the which-slit information was detected because the waves from the two slits just couldn't interfere with one another as they were entangled with different states of the counter. Only contributions to a wave in which all the remaining degrees of freedom are found in the same state may interfere.

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Thank you! "In principle, the state of the counters before their destruction inevitably leaves some traces in the environment because the information is preserved" Is it also true that the traces you mentioned can always be detected (in principle) from outside without opening the box, too? –  Eric Jun 27 '12 at 14:09
    
Dear Eric, it depends on how "perfectly" sealed the box is. ;-) If it is a box that in principle doesn't allow any waves to be transmitted, you can't see anything from the outside. ;-) But yes, one could argue that there can't exist such perfect boxes in Nature. At any rate, those things are really speculative because the procedures needed to decode the information are really undoable in practice. –  Luboš Motl Jun 28 '12 at 13:27
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An interesting special case:

Suppose the experiment consists of Bob at hole 2 in the double slit experiment being able to open and close the hole instantly. Let the intensity be so low that on average only one particle at a time is in the apparatus. By closing the hole he ensures the particle must go by route 1 if it is to hit the screen. Now what happens if he manages to re-open hole 2 just before the particle is detected at the screen? By repetition of the experiment the pattern built up at the screen is the interference pattern (case a). If the hole were closed at the time the particle is irreversibly detected at the screen then the pattern built up would not show interference at all (case b). The state of the apparatus at the exact moment of the irreversible detection of the particle at the screen determines if the particle is contributing to case a or case b pattern. Note that in this experiment Bob hasn't detected any particles himself but keeps a record of the time at which he opens or closes the hole which can be correlated with the particle arrival times at the screen. we can thus group the screen observations into two groups- those that occurred with the hole open and those with it closed. The first ones show the interference pattern the second do not. What happens if Bobs records are destroyed before the screen results are analyzed into these two groups? We see a mixed pattern of both a and b so interference fringes on top of high background which tends to wash out the fringes. The point is destroying Bobs information doesn't change the results which were observed at the screen. But those results were determined by the now lost information, they don't suddenly change.

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thank you for the explanation. –  Eric Oct 20 '13 at 13:21
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