“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.