## September 27, 2007

### A quantum measurement paradox: the reallocation by measurement problem

I spend lots of time thinking about puzzling foundational issues in quantum mechanics (search for "quantum measurement paradox" and I often show in the top five), and do I have a deal for anyone else like me. I have been thinking of this problem for awhile, and the post on Uncertain Principles stimulated me to post a version of it. If you want something really strange and "after the fact" about measurement, please consider my following proposal: In a Mach-Zehnder interferometer, insert a gray filter G into leg L2. Given a traditional stream of light, that alters the amplitudes delivered to the beam-splitter/recombiner R. With a 25% transmitting filter, the L2 amplitude is 0.5 of that in L1 (which itself is sqrt (0.5) of the original pre-split input.) Hence, with symmetrical R, we get an output mix rather than all A channel output. We can adjust R to a compensatory split so that output is again all A channel. Using individual photons, the statistics should be the same. FEL optical physicist Michelle Shinn of J-Lab agreed with me that's so, despite the weirdness of the photon's wave function in L2 being attenuated by the chance that it could have been absorbed, even if it wasn't (well, superposition of absorption and not-absorption in the dye molecules in the filter, etc, right?) Also, as G gets darker, this has to be the limiting factor approaching the results of an opaque stop in L2. But what happens if we can find out whether a photon has been absorbed in the filter?

Consider an opaque stop: the stop clearly "reallocates" the WF all into L1, in a manner akin to the Renninger negative result problem, even though no actual "measurement" is takeną„¤ But there, a photon will just never get through. However, G may or may not absorb a photon, something we can in principle check on (There are semi-transparent optical detectors, no? Just consider film for example.) Now, while G is still "deciding" (in a state of superposition) whether it will absorb or not, it makes sense to consider the L2 wave to be attenuated relative to L1. Maybe that's the normal time scale to allow interference in R before that happens. But, after a certain time, if we check G carefully to look for evidence of absorption, it should be settled: absorption or not. If it did, there's no paradox. But if we find "no absorption," why in the world should the L2 wave continue attenuated? The measurement result was "no" for G, so there is no longer "a chance" that the photon might end up there. The filter might as well have been clear glass, right? If so, then the interference at R would be different (it would follow normal equal-balance rules instead.)

The really weird thing is, that reallocation should take place as soon as the absorption/detection issue is settled. If so, we could manipulate the pattern of hits (with sequential photon shots) at the output by looking for evidence of absorption in the filter, which would start rearranging the WF as per Renninger etc. In principle, there's nothing to stop this from being a true FTL signal, since manipulating G (or perhaps the distance to R) causes noticeable effects (not distant signal correlations) at R. Sure, that's problematical, but you can't just blow off the supposed effect on the WF of the negative measurement in G, can you? Have fun.
(I also just put this up on sci.optics, sci.physics, etc.)