Kochenspek: Kochen–Specker Theorem
Quantum Foundations arc, page 11 of 193. Back to The Pixelated Physics Universe
Who they are
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.
Kochenspek is a character of the Quantum Foundations arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
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.
$$ \begin{gathered} \text{18 vectors, 9 orthogonal bases, each vector in 2 bases (4 dimensions)} \\ 9 \times 1 = 9 \ (\text{odd}) \quad \text{vs} \quad 2 \times (\text{number of vectors labelled } 1) \ (\text{even}) \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $\text{basis}$ | a set of 4 mutually orthogonal vectors in 4 dimensions | 1 |
| $0,\ 1$ | the answer assigned to each vector (exactly one 1 per basis) | 1 |
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.
Sources
- Kochen & Specker (1967) The Problem of Hidden Variables in Quantum Mechanics, J. Math. Mech. 17, 59
- Cabello, Estebaranz & García-Alcaine (1996) Bell-Kochen-Specker theorem: A proof with 18 vectors, Phys. Lett. A 212, 183
Crossovers
Kochenspek meets Reflexia (Nonlocal Mirror / Quantum Nonlocality). Reflexia rules out local hidden variables. Kochenspek rules out non-contextual ones, with no assumption about locality. Two different theorems close two different doors, and contextual, nonlocal theories such as Bohmian mechanics walk through neither.
Kochenspek meets Triumvira (GHZ State). Triumvira's three-qubit test and Kochenspek's coloured vectors attack the same idea: that every measurement result is fixed in advance. Each finds a set of quantum predictions that no fixed list of answers can reproduce.
Kochenspek meets Wignera (Wigner's Friend). Wignera asks whose measurement result counts as fact. Kochenspek adds that the results cannot all be pre-written, independent of which other measurements go with them. Together they make the observer's role harder to ignore, though neither theorem picks an interpretation.
Wear the lore
Concept keywords: Kochen–Specker Theorem, Kochenspek, Quantum Foundations, physics lore, Pixelated Physics