This website is dedicated the search for the smallest Kochen-Specker system. See the section The experiment below, for an explanation what Kochen-Specker systems are and why they are interesting. At the moment, this website primarily lists the results from a paper by Uijlen and Westerbaan. Please feel free to contribute: send an e-mail or contribute directly on github.
The experiment
Consider the following experiment. Measure the SPIN of a spin 1 particle along a direction. Or more concretely: shoot a deuterium atom through a certain fixed inhomogeneous magnetic field. Depending on the direction it was shot through the field, the particle will move undisturbed or deviate.
Quantum Mechanics only predicts the probability, given the direction, whether the particle will deviate. Its probabilistic prediction has been tested thoroughly. One wonders: is there a deterministic theory predicting the outcome of this experiment?
Kochen and Specker have shown that any such deterministic theory must be odd: it cannot satisfy the plausible (non contextual) SPIN axiom, that is: in exactly two out of three pairwise orthogonal directions the particle will deviate.
They have proven this by giving 117 points on the sphere, representing 117 directions, for which there can be no special {0,1}-valued coloring (called a 010-coloring), which corresponds to the predictions of a deterministc theory.
After Kochen and Speckers original paper, several people have tried to give a smaller set of points to demonstrate this fact. Such a finite set of points is called a Kochen-Specker system.
Smallest Kochen-Specker system
The smallest known Kochen-Specker system is due to Conway. It has 31 points and is shown on the right. We also know that the smallest Kochen-Specker system must have at least 24 points. A short history:
| by | year | bound |
|---|---|---|
| Kochen and Specker | 1975 | ≤ 117 |
| Penrose, Peres (independently) | 1991 | ≤ 33 |
| Conway | ~1995 | ≤ 31 |
| Li, Bright and Ganesh | 2023 | ≥ 24 |
| Li, Bright and Ganesh | 2022 | ≥ 23 |
| Uijlen and Westerbaan | 2016 | ≥ 22 |
| Uijlen and Westerbaan | 2014 | ≥ 21 |
| Arends, Ouaknine and Wampler | 2009 | ≥ 18 |
The lower bounds
Arends, Ouaknine and Wampler observed that a n point Kochen-Specker system corresponds to a certain n vertex graph: a graph that is both (sphere) embeddable and not 010-colorable. They have shown there are no such graphs on less than 18 vertices. Uijlen and Westerbaan demonstrated that there is also no such graph on less than 21 vertices.