The Kochen-Specker Theorem

I will now take a break from the prior series, and discuss a few fundamental results in quantum mechanics which were not given proper attention in the past on this blog. Today I am discussing the Kochen-Specker theorem, which rivals in importance Bell’s theorem.

When you read about this result the first time, it looks a bit dry and abstract, but in fact it is child’s play because it is nothing more than a coloring game. Originally the first proof was quite intricate, but later on, the late Asher Peres found a great simplification and I will discus this instead.

Before starting, we need one preliminary result: for particles of total spin 1, we can measure the square of the component of spin in a direction and get +1 or 0. So far nothing special, but quantum mechanics shows that if we perform such a measurement on three orthogonal directions (say on x, y, z) we will get two results of +1 and one result of 0. We do not know what result will be on which direction, but we will always get a zero result and two +1 results in some order. I’ll not prove this, but I want to look at its meaning instead.

We know that in quantum mechanics the results of experiments do not exist before measurement, but can we create a model which will recover the 1,1,0 prediction for spin one particles? This will work for three orthogonal directions, but what if are adding additional orthogonal directions? If particles do have definite properties before measurement, it should be possible in principle to pre-assign the values of +1 and 0 to all our measurement directions in such a way that the +1,+1, 0 theorem is obeyed. The Kochen-Specker theorem shows that this is impossible.

Let us start with the measurement directions that Peres found:

(I am adapting this from a famous paper about Free Will: http://arxiv.org/pdf/quant-ph/0604079v1.pdf )

So here is the explanation: start with a cube, inscribe a circle on each face and add a point on the 4 places it touches the face sides (e.g. points V,D, U, etc). Then add a point in the middle of each face (points X, Y, Z) and connect this with the vertexes of the square. Add a point where this line intersects the circle (C, B, C’, B’) and unite them in a smaller square. Draw perpendiculars from the center of the face to the small square and add 4 more points (e.g. D, D’). Finally connect the center of the cube with all those points and obtain 33 directions: 13*3 – 4*3/2 = 33 directions [13 point on a face*3 faces but all 4 points on the inscribed circle are counted twice]

Now we can start the coloring game and prove the Kochen-Specker theorem. We will color the 1’s as red, and the 0’s as blue and see if this can be done in general or not.

Step 1: X, Y, Z for 3 orthogonal directions:

(we can pick X as zero without loss of generality)

Step 2: X-A implies A = +1 because the center of the cube and X and A form orthogonal directions, and on any 3 orthogonal directions there can be only one zero which is at X now) In turn A’ = 1 because X, A, A’ form 3 orthogonal directions.

Step 3: A,B,C form 3 orthogonal directions (this shows the cleverness of Peres’ choice for directions, try to prove using simple geometry that A,B,C form 3 orthogonal directions). Without loss of generality we can pick B=1, C=0.

and from A’ B’ C’: B’ = 1 and A’ = 1

Step 4: orthogonality of CD implies D = 1. Similarly C’D’ implies D’ = 1

Step 5: Z, D, E orthogonality implies E = 0. Z, D’, E’ implies E’ = 0

Step 6: EF and EG orthogonality implies G=G = 1. E’F’ and E’ G’ implies F’ = G’ = 1

Step 7: F, F’, U implies U = 0

Step 8: G, G’, V implies V = 0

Now for the contradiction: U is orthogonal with V and you cannot have both of them equal with zero.

So what does this mean? This shows that we cannot have a context independent assignment of measurement outcome before measurement. If we want to pre-assign measurement properties, this can only be done within the context of the measurement setting. K-S theorem weakens the idea of objective reality independent of measurement.

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