Quantum mechanics forces us to refine our attitude to counterfactual conditionals: questions about what would have happened if we had done something, even though we didn’t.
“What would the position of the particle be if I’d measured that… when actually I measured its momentum?” Here you’ll usually get no definite answer.
But sometimes you can use quantum mechanics to find out what would have happened if you’d done something… when classically it seems impossible!
Suppose you have a bunch of bombs. Some have a sensor that will absorb a photon you shine on it, and make the bomb explode! Others have a broken sensor, that won’t interact with the photon at all.
Can you choose some working bombs? You can tell if a bomb works by shining a photon on it. But if it works, it blows up—and then it doesn’t work anymore!
So, it sounds impossible. But with quantum mechanics you can do it. You can find some bombs that would have exploded if you had shone photons at them!
Here’s how:
Put a light that emits a single photon at A. Have the photon hit the half-silvered mirror at lower left, so it has a 50% chance of going through to the right, and a 50% chance of reflecting and going up. But in quantum mechanics, it sort of does both!
Put a bomb at B. Recombine the photon’s paths using two more mirrors. Have the two paths meet at a second half-silvered mirror at upper right. You can make it so that if the bomb doesn’t work, the photon interferes with itself and definitely goes to C, not D.
But if the bomb works, it absorbs the photon and explodes unless the photon takes the top route… in which case, when it hits the second half-silvered mirror, it has a 50% chance of going to C and a 50% chance of going to D.
So:
• If the bomb doesn’t work, the photon has a 100% chance of going to C.
• If the bomb works, there’s a 50% chance that it absorbs the photon and explodes. There’s also a 50% chance that the bomb does not explode—and then the photon is equally likely to go to either C or D. So, the photon has a 25% chance of reaching C and a 25% chance of reaching D.
So: if you see a photon at D, you know you have a working bomb… but the bomb has not exploded!
For each working bomb there’s:
• a 50% chance that it explodes,
• a 25% chance that it doesn’t explode but you can’t tell if it works,
• a 25% chance that it doesn’t explode but you can tell that it works.
This is the Elitzur–Vaidman bomb-testing method. It was invented by Avshalom Elitzur and Lev Vaidman in 1993. One year later, physicists actually did an experiment to show this idea works… but alas, not using actual bombs!
In 1996, Kwiat showed that using more clever methods, you can reduce the percentage of wasted working bombs as close to zero as you like. And pushing the idea even further, Graeme Mitchison and Richard Jozsa showed in 1999 that you can get a quantum computer to do a calculation for you without even turning it on!
This sounds amazing, but it’s really no more amazing than the bomb-testing method I’ve already described.
References
For details, read these:
• A. Elitzur and L. Vaidman, Quantum mechanical interaction-free measurements, Found. Phys. 23 (1993), 987–997.
• Paul G. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M. Kasevich, Experimental realization of “interaction-free” measurements.
• Paul G. Kwiat, Interaction-free measurements.
• Graeme Mitchison and Richard Jozsa, Counterfactual computation, Proc. Roy. Soc. Lond. A457 (2001), 1175–1194.
The picture is from the Wikipedia article, which also has other references:
• Elitzur–Vaidman bomb tester, Wikipedia.
Bas Spitters pointed out this category-theoretic analysis of the issue:
• Robert Furber and Bart Jacobs, Towards a categorical account of conditional probability.
Azimuth 
OK this is getting out of hand:-) Apologies if this is answered in the references-
What’s described seems to be a device that, when the photon takes the top route and ends up at D, determines the position of something (namely, a photon absorber) to a fixed uncertainty without disturbing that thing’s momentum at all.
Is that right, and if so why can’t two such devices, configured at right angles, be used to violate the position-momentum version of the uncertainty principle?
“two such devices, positioned at right angles” (if I understand what you mean by that) would just measure two position coordinates at once (when both photons took the proper paths for that), and still not measure momentum at all.
Thanks for trying to tackle arch1’s question. I didn’t understand it, so I just moved on to some other more pressing chore.
Thanks Bruce.
I think that if the photon absorber were determined to be on both of two intersecting paths, this would actually determine all three of its position coordinates.
As for the momentum measurement – er, I thought *you* were handling that part:-) (Actually my Q was motivated by a vaguely remembered justification of the UP which depended on an interrogating photon’s disturbing the measured particle; if the disturbance went away, I muddily reasoned, so would the explanation:-)
I agree, it could determine all three position coordinates to some fixed accuracy. Still no paradox, since momentum of photon absorber doesn’t matter for result, thus doesn’t get measured in any way.
It’s probably worthwhile trying to get some sort of paradox to arise from this strange phenomenon. You won’t succeed, since quantum mechanics is consistent, but you may turn up interesting ideas in this process, or at least improve your understanding of quantum mechanics.
Yes it will be fun and maybe instructive to mull over. Thanks John.
Even weirder is counterfactual quantum computing. As if QC wasn’t weird enough already, you can get an answer without actually running it !!
Click to access Hosten-Nature-439-949-952.pdf
What if there were another kind of broken bomb, which *does* absorb the photon but *doesn’t* explode when it absorbs it. Let’s call it a “fake bomb”. As far as I can see from what’s explained here, this scheme can’t help you distinguish fake bombs from working bombs. Is that right? That would be much more useful in practice — is there any scheme which could conceptually do that? (E.g. trying to send another photon through the hypothetical, but presumed reliably present when it should be, debris cloud from the explosion?)
Bruce wrote:
Right. Stripped of all its drama, this scheme just tells, a certain fraction of the time, whether or not a photon going through one of the two paths would have been absorbed.
Using some math you could abstract this idea down to its bare conceptual bones. Then the fun part would be to generalize it and consider lots of other ‘counterfactual procedures’.
But I don’t have the energy to do this now, or apply it to the detection of fake bombs!
You might enjoy reading Scott Aaronson’s blog post which attempts to pour cold water on some of the more overheated reactions to Graeme Mitchison and Richard Jozsa’s paper Counterfactual computation.
Over on G+, Jeff Lundeen wrote:
[…] (merely watching an atom stops it from decaying) and the ultimate in quantum bizarreness, the Elitzur-Vaidman Bomb Test, where it is possible to detect the presence of a bomb using a photon that doesn’t go anywhere […]
This is a fantastic explanation, thank you. One part I still don’t get: “You can make it so that if the bomb doesn’t work, the photon interferes with itself and definitely goes to C, not D.” Does the self-interference cause the wave function to collapse? If so, why wouldn’t the particle ever go to D? Thanks again!
Self-interference does not cause the wavefunction to collapse. Self-interference can make the wavefunction cancel out in certain locations, making it equal zero, making the probability of certain events zero.