Measurement without interaction in quantum mechanics

In front of you is a sealed box, which either contains nothing OR an incredibly powerful nuclear bomb, the explosion of which threatens to wipe out humanity permanently. Even worse, this bomb is incredibly unstable and will blow up at the slightest contact with a single photon. This means that anybody that opens the box to look inside and see if there really is a bomb in there would end up certainly activating it and destroying the world. We don’t have any way to deactivate the bomb, but we could maintain it in isolation for arbitrarily long, despite the prohibitive costs of totally sealing it off from all contact.

Now, for obvious reasons, it would be extremely useful to know whether or not the bomb is actually active. If it’s not, the world can breathe a sigh of relief and not worry about spending lots of money on keeping it sealed away. And if it is, we know that the money is worth spending.

The obvious problem is that any attempt to test whether there is a bomb inside will involve in some way interacting with the box’s contents. And as we know, any such interaction will cause the bomb to detonate! So it seems that we’re stuck in this unfortunate situation where we have to act in ignorance of the full details of the situation. Right?

Well, it turns out that there’s a clever way that you can use quantum mechanics to do an “interaction-free measurement” that extracts some information from the system without causing the bomb to explode!

To explain this quantum bomb tester, we have to first start with a simpler system, a classic quantum interferometer setup:

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At the start, a photon is fired from the laser on the left. This photon then hits a beam splitter, which deflects the path of the photon with probability 50% and otherwise does nothing. It turns out that a photon that gets deflected by the beam splitter will pick up a 90º phase, which corresponds to multiplying the state vector by exp(iπ/2) = i. Each path is then redirected to another beam splitter, and then detectors are aligned across the two possible trajectories.

What do we get? Well, let’s just go through the calculation:

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We get destructive interference, which results in all photons arriving at detector B.

Now, what happens if you add a detector along one of the two paths? It turns out that the interference vanishes, and you find half the photons at detector A and the other half at detector B! That’s pretty weird… the observed frequencies appear to depend on whether or not you look at which path the photon went on. But that’s not quite right, because it turns out that you still get the 50/50 statistics whenever you place anything along one path whose state is changed by the passing photon!

Huh, that’s interesting… it indicates that by just looking for a photon at detector A, we can get evidence as to whether or not something interacted with the photon on the way to the detector! If we see a photon show up at the detector, then we know that there must have been some device which changed in state along the bottom path. Maybe you can already see where we’re going with this…

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We have to put the box in the bottom path in such a way that if the box is empty, then when the photon passes by, nothing will change about either its state or the state of the photon. And if the box contains the bomb, then it will function like a detector (where the detection corresponds to whether or not the bomb explodes)!

Now, assuming that the box is empty, we get the same result as above. Let’s calculate the result we get assuming that the box contains the bomb:

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Something really cool happens here! We find that if the bomb is active, there is a 25% chance that the photon arrives as A without the bomb exploding. And remember, the photon arriving at detector A allows us to conclude with certainty that the bomb is active! In other words, this setup gives us a 25% chance of safely extracting the information about if the bomb is active!

25% is not that good, you might object. But it sure is better than 0%! And in fact, it turns out that you can strengthen this result, using a more complicated interferometer setup to learn with certainty whether the bomb is active with an arbitrarily small chance of setting off the bomb!

There’s so many weird little things about quantum mechanics that defy our classical intuitions, and this “interaction-free measurements” is one of my new favorites.

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