Kochen–Specker Theorem

The Kochen–Specker theorem rules out a global noncontextual assignment of predetermined values to all sharp quantum events in complex Hilbert-space dimension at least three. The canonical Kochen–Specker Theorem page owns the Gleason-based proof, finite parity witness, qubit exception, and comparison with generalized contextuality.

Helpful background. For the projector proof, Gleason corollary, and finite parity witness behind this lookup card, consult the canonical Kochen–Specker Theorem page.

Projector statement

Let $\mathcal P_1(\mathcal H)$ be the rank-one projectors on a finite-dimensional complex Hilbert space with $d\ge3$. There is no map

$$v:\mathcal P_1(\mathcal H)\longrightarrow\{0,1\}$$

such that every orthonormal basis with projectors $P_1,\ldots,P_d$ satisfies

$$\sum_{j=1}^{d}v(P_j)=1.$$

The same projector must retain its value in every compatible basis in which it appears. Functional relations among commuting observables are preserved.

Scope

The contradiction is global: one basis can always be colored, but overlapping contexts cannot all be colored consistently. The standard rank-one projector theorem requires $d\ge3$; traditional qubit projectors admit colorings. This does not exempt qubits from generalized operational contextuality, which uses broader preparation, transformation, and measurement equivalences.

The theorem excludes noncontextual value models of the specified form. It does not exclude contextual hidden-variable theories, prove that measurement disturbance alone is contextuality, or coincide with Bell’s spatial locality theorem.

Relation to Gleason

Gleason’s Theorem gives every normalized noncontextual projector probability the trace-rule form $\mu(P)=\operatorname{Tr}(\rho P)$. Kochen–Specker is the dispersion-free obstruction: no such global assignment can take only the values $0$ and $1$.

Quick check

Why does coloring a single orthonormal basis never prove the theorem?

Solution

Inside one basis, choose one projector to have value $1$ and all others value $0$. The contradiction requires multiple overlapping bases, because noncontextuality forces a shared projector to retain one value wherever it occurs.

References

• J. S. Bell, “On the problem of hidden variables in quantum mechanics,” Reviews of Modern Physics 38, 447–452 (1966).
• S. Kochen and E. P. Specker, “The Problem of Hidden Variables in Quantum Mechanics,” Indiana University Mathematics Journal 17, 59–87 (1968), doi:10.1512/iumj.1968.17.17004.
• N. D. Mermin, “Simple unified form for the major no-hidden-variables theorems,” Physical Review Letters 65, 3373–3376 (1990).