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The source S sends a photon into the beam splitter below. There is a 50% chance that it will be detected at A and a 50% chance it will be detected at B.

S -----------\---------> A
             |
             |
             |
             v
             B

Now if we assume that physics is (generally) reversible we should be able to time-reverse this process. This would imply that if we send in a photon at A or B it should always appear back at S.

How can this happen? We know that if we send in a photon at A (or B) we would expect it to appear 50% of the time at S and 50% at C as below.

             C                                         C
             ^                                         ^
             |                                         |
             |                                         |
             |                                         |
S <----------\--------- A               S <------------\
                                                       |
                                                       |
                                                       |
                                                       B

In order that the photon always appears back at S and never at C we need a superposition of the two situations above such that the paths to S constructively interfere and the paths to C destructively interfere.

Thus it seems that in order to retain time-reversibility we need to assume a many-worlds view. As a photon is sent in at A (or B) it must be assumed that another photon is simultaneously sent in at B (or A) and both states weighted with the correct amplitude such that a photon appears with certainty back at S.

What do people think?

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Time-reversal symmetry is a property of quantum mechanics. All interpretations of quantum mechanics are supposed to be consistent with quantum mechanics, so all interpretations of quantum mechanics have to be consistent with time-reversal symmetry. This includes the Copenhagen interpretation (CI). If a particular way of expressing the CI appears to violate time-reversal symmetry, then that's a problem with the awkwardness of that particular statement of the CI.

"Thus it seems that in order to retain time-reversibility we need to assume a many-worlds view. As a photon is sent in at A (or B) it must be assumed that another photon is simultaneously sent in at B (or A) and both states weighted with the correct amplitude such that a photon appears with certainty back at S."

I don't think the logical link between the first and second sentences quite works.

Although you don't say it explicitly, I think your reasoning is that (1) detectors or observers at A and B have to be in a superposition of states, like Schrodinger's cat, correlated with one another, and in a coherent state; and therefore (2) the many-worlds interpretation (MWI) must hold. First off, note that none of this has anything to do with time-reversal. If #1 holds in the time-reversed version, it holds in the original version. If it holds in the original version, it holds in the time-reversed one. I also don't think #1 implies #2. I doubt that #1 is true. If #1 is true, it doesn't disprove CI. If #1 is false, it doesn't disprove MWI.

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