The Kochen-Specker theorem states that quantum observables in a complex Hilbert space of dimension at least 3 cannot all be assigned predetermined values consistently with the algebraic relations between compatible observables and independently of which compatible observables are measured together (Kochen and Specker 1967, Budroni et al. 2022). This failure is called quantum contextuality.
One formulation considers the rank-one orthogonal projections in a Hilbert space 
. There is no function 
from these projections
to 
such that, for every orthonormal basis 
of 
,
 |
where 
is the orthogonal projection onto the span of 
. Equivalently, the rays of 
cannot be colored 0 and 1 so that every
orthonormal basis contains exactly one ray
colored 1. Finite sets of rays already force this contradiction.
The Peres-Mermin square and Mermin pentagram give especially short state-independent parity proofs in dimensions 4 and 8, respectively. Their observables are elements of Pauli groups organized into commuting contexts. More generally, these observable-based proofs can be encoded by hypergrams whose vertices are observables, whose hyperedges are contexts, and whose graph edges represent anticommutation.
