In physics, the Elitzur-Vaidman bomb-testing problem is a thought experiment in quantum mechanics, first proposed by Avshalom Elitzur and Lev Vaidman in 1993. An actual bomb-tester was constructed and successfully tested by Anton Zeilinger, Paul Kwiat, Harald Weinfurter, and Thomas Herzog in 1994.^[1] It employs a Mach-Zehnder interferometer for ascertaining whether a measurement has taken place.

The following example illustrates the bomb-testing problem: Consider a collection of bombs, some of which are duds. The bombs are triggered by a single photon. Unstable bombs will absorb the photon and detonate. Dud bombs will not absorb the photon. The problem is how to separate the usable bombs from the duds. A bomb sorter could accumulate dud bombs by attempting to detonate each one. Unfortunately, this process destroys all the unstable bombs.

Now consider a slight variation: a mirror attached to a plunger, which is attached to the detonator. A photon that impinges on the mirror pushes the plunger and this detonates the bomb. The duds are those bombs whose "plunger" is stuck, so that even if a photon hits the mirror, the plunger does not get pushed, and no detonation occurs. This property is important, because it means a dud effectively just reflects the photon. In the good bomb scenario, the photon never actually hits the bomb's mirror—but this is enough to know that the photon went via the other path (a "null" measurement).

A solution is for the sorter to use a mode of observation known as counterfactual measurement, which relies on principles of quantum mechanics.^[2]

Start with a Mach-Zehnder interferometer and a light source which emits single photons. When a photon emitted by the light source reaches a half-silvered plane mirror, it has equal chances of passing through or reflecting.^[3] On one path, place a bomb (B) for the photon to encounter. If the bomb is working, then the photon is absorbed and triggers the bomb. If the bomb is non-functional, the photon will pass through the dud bomb unaffected.

When a photon's state is non-deterministically altered, such as interacting with a half-silvered mirror where it non-deterministically passes through or is reflected, the photon undergoes quantum superposition, whereby it takes on all possible states and can interact with itself (entangled photons). This phenomenon continues until an observer interacts with it, causing the wave function to collapse and returning the photon to a deterministic state.

A step-by-step explanation of what happens:

If the bomb is a dud:

• The photon both (i) passes through the 1st half-silvered mirror and (ii) is reflected.
• The bomb will not absorb a photon, and so the lower route is a possible path to the point of interference.
• The system reduces to the basic Mach-Zehnder apparatus with no sample, in which constructive interference occurs along the path horizontally exiting towards (D) and destructive interference occurs along the path vertically exiting towards (C).
• Therefore, the detector at (D) will detect a photon, and the detector at (C) will not.

If the bomb is usable:

• The photon both (i) passes through the 1st half-silvered mirror and (ii) is reflected.
• Upon meeting the observer (the bomb), the wave function collapses and the photon must be either on the lower route or on the upper route, but not both.
• If the photon actually takes the lower route:
  • Because the bomb is usable, this photon triggers the bomb and it explodes.
• If the photon actually takes the upper route:
  • Since the lower route is not taken, there will be no interference effect at the 2nd half-silvered mirror.
  • The photon on the upper route now both (i) passes through the 2nd half-silvered mirror and (ii) is reflected.
  • Upon meeting further observers (detector C and D), the wave function collapses again and the photon must be either at detector C or at detector D, but not both.

Therefore, there are only three observable results:

1. The bomb explodes.
2. The bomb does not explode and only detector (C) detects the photon. The bomb must be usable.
3. The bomb does not explode and only detector (D) detects the photon. It is possible that the bomb is usable or that it is a dud.

In the case of the third observation, the experiment may be repeated to see if the bomb will explode or if detector (C) will detect a photon. On average, this will identify all of the dud bombs, explode two thirds of the usable bombs, and identify one third of the usable bombs without detonating them.

An easy way to understand the experiment is to use the practical common knowledge about choosing we all use in daily life. Each entangled photon corresponds to an alternative. The collapse of the superposition of the entangled photons corresponds to a decision on the alternatives. The point of interference corresponds to the alternatives coming out equal. So, in common language, what this experiment tries to find out, is if the photon could have been on the path in between the bomb (live or dud) and the second half silvered mirror. If there is a live bomb then this possibility does not exist, but if it is a dud then this possibility is real. The experiment looks in retrospect whether or not this particular alternative did exist. Same like as in daily life, when you know you had an alternative thing you could have done, and know that alternative was real, eventhough you chose not to do that.

In 1994, Anton Zeilinger, Paul Kwiat, Harald Weinfurter, and Thomas Herzog actually performed an equivalent of the above experiment, proving interaction-free measurements are indeed possible.^[1]

In 1996, Kwiat et al. devised a method, using a sequence of polarising devices, that efficiently increases the yield rate to a level arbitrarily close to one. The key idea is to split a fraction of the photon beam into a large number of beams of very small amplitude, and reflect all of them off the mirror, recombining them with the original beam afterwards.^[4] ( See also http://www.nature.com/nature/journal/v439/n7079/full/nature04523.html#B1 .) It can also be argued that this revised construction is simply equivalent to a resonant cavity and the result looks much less shocking in this language.

This experiment is philosophically significant because it determines the answer to a counterfactual question: "What would happen were the photon to pass through the bomb sensor?". The answer is either: "the bomb works, the photon was observed, and the bomb will explode", or "the bomb is a dud, the photon was not observed, and the photon passes through unimpeded".

If we were actually to perform the measurement, any bomb would actually explode. But here the answer to the question "what would happen" is determined without the bomb going off. This provides an example of an experimental method to answer a counterfactual question.

[edit] See also

• Counterfactual definiteness
• Interaction-free measurement

[edit] References

• P. G. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M. A. Kasevich (1995). "Interaction-free Measurement". Phys. Rev. Lett. 74: 4763. doi:10.1103/PhysRevLett.74.4763. 
• Paul G. Kwiat; H. Weinfurter, T. Herzog, A. Zeilinger, and M. Kasevich. "Experimental realization of "interaction-free" measurements" (pdf). http://www.quantum.univie.ac.at/publications/pdffiles/1994-08.pdf. Retrieved 2007-12-08. 
• Paul G. Kwiat. "Tao of Interaction-Free Measurements". Archived from the original on 1999-02-21. http://web.archive.org/web/19990222174102/www.p23.lanl.gov/Quantum/kwiat/ifm-folder/ifmtext.html. Retrieved 2007-12-08. 
• Paul Kwiat. "Current Location of "Tao of Interaction-Free Measurements"". http://physics.illinois.edu/people/kwiat/interaction-free-measurements.asp. Retrieved 2009-04-01. 
• Keith Bowden (1997-03-15). "Can Schrodinger's Cat Collapse the Wavefunction?". http://nonlocal.com/quantum-d/v2/kbowden_03-15-97.html. Retrieved 2007-12-08. 
• David M. Harrison (2005-08-17). "Mach-Zehnder Interferometer". http://www.upscale.utoronto.ca/GeneralInterest/Harrison/MachZehnder/MachZehnder.html. Retrieved 2007-12-08. 
• Elitzur A. C. and Vaidman L. (1993). Quantum mechanical interaction-free measurements. Found. Phys. 23, 987-97.
• Penrose, R. (2004). The Road to Reality: A Complete Guide to the Laws of Physics. Jonathan Cape, London.
• G.S. Paraoanu (2006). "Interaction-free Measurement". Phys. Rev. Lett. 97: 180406. doi:10.1103/PhysRevLett.97.180406.