The setup consists of two beam splitters BS₁ and BS₂, two mirrors M₁ and M₂, two perfect (100% efficient) photodetectors D₁ and D₂, and (maybe, maybe not) a bomb so sensitive that the detection of a single photon causes it to explode.

Bomb testing

A photon enters BS₁ from the left. If the bomb is present, the route taken by the photon is measured: either the bomb explodes, in which case the photon has taken the upper route, or the bomb does not explode, in which case the photon has taken the lower route. BS₁ is designed so that a photon detected by either D₁ or D₂ has taken either the upper or the lower route with probability 1/2. If it takes the route via M₂, either photodetector clicks with probability 1/2. Hence if the bomb is present, the following statistics is observed:

If the bomb is absent, Rule B applies. To be able to calculate the amplitudes associated with the alternatives, all we need to know is that the absolute values of the amplitudes are equal, and that their phases are equal except that they contain a factor e^iπ/2 = i for every reflection. (The magnitude of the phase shift depends on the material. For the sake of convenience we use mirrors and beam splitters for which it equals π/2.)

Since both alternatives leading to D₁ involve two reflections, the corresponding amplitudes are equal. The probability of detection by D₁ is the absolute square of the sum of these amplitudes: |A+A|² = 4|A|².

Under the conditions stipulated by Rule A the probability of detection by D₁ is |A|²+|A|² = 2|A|². Observe that if there's no way to ascertain the route taken by the photon, the photon is twice as likely to reach D₁ than if there is a way.

The alternative leading to D₂ via M₁ involves three reflections. The corresponding amplitude therefore contains the extra factor i³ = -i. The alternative via M₂ involves a single reflection, so its amplitude contains the extra factor +i. The probability of detection by D₂ is therefore |-iA+iA|² = 0. Thus if the bomb is absent, the following statistics is observed:

It is therefore possible to learn whether the bomb is present without setting it off, albeit with a success rate of only 25%. If the bomb is present, there is a 50% chance that it will explode, a 25% chance that the photon will be detected by D₁, and a 25% chance that it will be detected by D₂. If it is detected by D₁, we learn nothing, for D₁ can click in either case. But if the photon is detected by D₂, then we have learned that the bomb is present without setting it off, for D₂ would not have clicked if the bomb had been absent.

When a version of this experiment was demonstrated at a science fair in Groningen (Holland) in 1995, the reactions of non-physicists differed markedly from those of physicists. Everyone was perplexed, for the detection of the photon by D₂ implies that

If the photon never came near the bomb, how can it possibly tell us that the bomb was present?

Interestingly, while the ordinary folks were of the opinion that some physicist would eventually figure this out, the physicists themselves though that this puzzle probably would never be solved.

Based on A. C. Elitzur and L. Vaidman, "Quantum mechanical interaction-free measurements," Foundations of Physics 23, 987-97, 1993.