Kochenspek

Kochenspek wears a crown of vectors and tries to paint each one 0 or 1. Every time, one basis is left without a consistent answer.

The Physics

Can every quantum measurement have a pre-set answer, waiting to be read, no matter which other compatible measurements are made alongside it? Kochen and Specker proved in 1967 that it cannot. In a Hilbert space of dimension 3 or more, you cannot assign 0 or 1 to every direction so that each orthogonal basis gets exactly one 1. Their original proof used 117 directions in three dimensions.

Cabello, Estebaranz and García-Alcaine gave a short proof in four dimensions: 18 vectors arranged in 9 orthogonal bases, with each vector in exactly 2 bases. A parity count finishes it.

9 bases × 1 one each = 9 (odd) but each vector counted twice ⇒ even

• Basis: a set of 4 mutually orthogonal vectors in 4 dimensions
• 9 bases × 4 vectors = 36 = 18 vectors × 2

Kochen & Specker, J. Math. Mech. 17, 59 (1967); Cabello, Estebaranz & García-Alcaine, Phys. Lett. A 212, 183 (1996).

Honest caveat: The theorem rules out non-contextual hidden variables, not all hidden variables: contextual theories such as Bohmian mechanics survive. It is not Bell's theorem and makes no locality assumption. The mathematics is settled; what it means for interpretations is still debated.

On the shirt

Front: Kochenspek, the pixel-art character, with “NO HIDDEN LABELS”. Back: a schematic diagram of 9 bases and 18 vectors (an illustration of the counting, not Cabello's actual vectors), the parity argument and the 1967 and 1996 sources. Printed on a unisex tee.