9.8 The Elitzur-Vaidman bomb problem
The problem and its solution were proposed by Avshalom Elitzur and Lev Vaidman in 19936. Consider a bomb with an ultra-sensitive detonator on its nose that even a photon impinging on the detonator’s slightly wobbly mirror can set it off (see Fig. 9.1). The bomb’s production line is not perfect—in some cases the detonator is jammed, and the bomb fails to explode and is classed a dud. The problem is to determine if a bomb’s plunger is stuck or not without exploding the bomb. Classical physics cannot determine without exploding the bomb. The solution is an interesting application of the Mach-Zehnder interferometer. The photon source emits a single photon. Now two possible paths for the photon exist.
If the photon takes the |h1〉 path (50% probability), the bomb’s mirror will either absorb the photon and wobble and the bomb will explode, or it will simply send the photon along the |v2〉 path because the plunger is stuck and the photon will be detected by the detector B. In this case 50% of good bombs will explode.
The more interesting case is when the photon takes the |v1〉 path (50% probability). The photon then does not impinge on the bomb’s mirror and so even a good bomb cannot explode. But the photon’s superposed state will still sense if the |h1〉 path has a wobbly mirror or not! If it senses a jammed mirror, the photon will end up at detector B. If it senses a wobbly mirror, it can end up at either detector A or at B. Note that if A detects a photon then the bomb is not a dud. Thus, in half of the cases where an active bomb does not explode will the detector A register a photon. At the end of the tests, we would have found only a quarter of the originally active bombs which are guaranteed to be actually still active. We can continue to repeat the tests on the remaining doubtful bombs till no doubtful bomb is left. Ultimately, we will obtain just one third (since 1/4 + 1/16 + 1/64 +… = 1/3) of the active bombs that we started with, but they are now guaranteed to be active.
Quantum advantage: Classically there would be no way of saving even a single good bomb, but quantum mechanically one can save one third of them. With some refinements, the two-thirds wastefulness can be reduced effectively to one-half (Elitzur and Vaidman, 1993)7. In 1995, Kwiat, Weinfurter, Herzog, and Zeilinger reported an experiment verifying the Elitzur-Vaidman result, thus proving that interaction-free measurements are indeed possible8. The same year, Kwiat, Weinfurter, Zeilinger, Herzog, and M. Kasevich devised a method, using a sequence of polarizing devices, which efficiently increased the yield rate to a level arbitrarily close to one9.
6 Elitzur & Vaidman (1993). See also: Penrose (1994), pp. 239-240 and 268-270;
and Baeyer (2001), p. 16.
7 Elitzur & Vaidman (1993).
8 Kwiat, et al (1995a). See also: Kwiat, et al (1995b).
9 Kwiat, et al (1995c). See also: DeWeerd (2002).