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Really short introduction to the Kochen-Specker paradox

The Kochen-Specker paradox is, in its simplest incarnation, talking about spin-1-particles. When measuring spin for such a particle along a direction, only three values are possible: parallel to the direction, orthogonal to it, and anti-parallel to it, mathematically denoted +1, 0, and -1. In quantum mechanics, simultaneous measurement of spin along three orthogonal axes is impossible, but measuring the square of the spin is possible. From quantum mechanics we also know that the sum of these spin squares must be 2 (in the spin-1 case).

In simple experimental setups, there is usually only one direction specified (the direction along which the spin component squared is measured). Thus, it is not unnatural to expect that in the more complicated triple-measurement setting, the measurement result along, say, the z-axis, is independent of what directions are used in the other measurements, given that they are in the x-y plane.

We thus have two properties of our measurement result:

  • Noncontextuality: A measurement result is not affected by the context of the measurement.
  • Quantum-mechanical results: Only the results 0 and 1 are possible, and the result when measuring on an orthonormal triad must sum up to 2.

The Kochen-Specker paradox is the statement that one cannot assign 0 or 1 to all directions (in regular 3D space). Even for some finite collections of directions there is a contradiction.

Example: This is a 33-vector set devised by Asher Peres, on which it is impossible to assign values in this way. It is only the direction that is important here, so I have only drawn dots where the directions hit the unit sphere. Green dots indicate the value 0 and red dots the value 1. The example goes like this:

[Peres image 1] One of the axes has to be green, so choose one. Orthogonal vectors are red.
[Peres image 2] Two directions on the 45° latitude are orthogonal, and since the equator is all red, one of these has to be green. Choose one. Orthogonal vectors are red.
	image 3] Same holds for these two. Choose one. Orthogonal vectors are red.
[Peres image 4] Here is another orthogonal pair on the 45° latitude, again the third vector is on the equator. Choose one green, orthogonal vectors are red. We have now a number of red vectors, some of which are orthogonal in such a way that they force two new vectors to be green (the checkered green ones in the figure).
[Peres image 5] Choose one of them to be green (we'll save the other for later), and this choice will force another vector to become red. This new red vector forces a new green one.
[Peres image 6] Color that green, we have another red, and another green.
[Peres image 7] Color that green, we have another red, and another green.
[Peres image 8] Color that green, we have another red, and another green.
[Peres image 9] Color that green, we have another red, and another green.
[Peres image 10] Color that green, we have another red, no wait, that one is forced to be green by another pair of red vectors! Contradiction.

The four choices we made at the top are not a restriction; the symmetry of the set of vectors implies that these choices can be made either way. This set cannot be colored in this way.

Jan-Åke Larsson

Jan-Åke Larsson


Department of Electrical Engineering
Linköping University

+46(0)13 281468

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Page responsible: Jan-Åke Larsson
Last updated: 2010-11-10