Contextuality

Bell nonlocality is a special case of a deeper phenomenon: contextuality. Where nonlocality exploits spatial separation, contextuality needs no separation at all — only that quantum measurements cannot all carry predetermined, context-independent values. The GHZ argument was secretly a contextuality proof, and this lesson makes that general.

Noncontextual hidden variables

A noncontextual hidden-variable (NCHV) model assigns to every observable $A$ a definite value $v(A)$ , fixed before measurement, with two consistency rules. Outcome rule: $v(A)$ is an eigenvalue of $A$ . Functional / product rule: for commuting observables, the assignment respects algebra — e.g. $v(AB)=v(A)v(B)$ and $v(A+B)=v(A)+v(B)$ whenever $[A,B]=0$ . "Noncontextual" means $v(A)$ does not depend on which other compatible observables are measured alongside $A$ (its "context"). Bell locality is just the special case where the contexts are enforced by distance.

The Kochen–Specker theorem

In any Hilbert space of dimension $d\ge 3$ , no NCHV assignment exists that is consistent for all observables. The Kochen–Specker (KS) theorem proves this with a finite set of projectors: one can exhibit a collection of measurement bases such that no $\{0,1\}$ value assignment picks exactly one "1" per basis while respecting shared vectors between bases. Quantum predictions cannot be reproduced by predetermined, context-independent outcomes — a state-independent no-go.

The Peres–Mermin square

The most transparent proof is the Peres–Mermin magic square, a $3\times3$ array of two-qubit observables:

\begin{array}{ccc} X\!\otimes\! I & I\!\otimes\! X & X\!\otimes\! X \\ I\!\otimes\! Z & Z\!\otimes\! I & Z\!\otimes\! Z \\ X\!\otimes\! Z & Z\!\otimes\! X & Y\!\otimes\! Y \end{array}

Every entry squares to $I$ (eigenvalues $\pm1$ ), and the three observables in each row and in each column mutually commute, so each line is jointly measurable. Computing operator products:

each of the three rows multiplies to $+I$ ,
the first two columns multiply to $+I$ , and the third column multiplies to $-I$ .

The contradiction

Assume an NCHV model assigns $v=\pm1$ to each of the nine entries. By the product rule applied to each commuting line, the product of the three values along every row equals $+1$ and along every column equals the operator product ( $+1, +1, -1$ ). Now multiply all nine values twice over — once grouped by rows, once by columns. Every entry appears exactly twice, so both groupings give the same number, $+1$ . But:

\underbrace{(+1)(+1)(+1)}_{\text{rows}} = +1 \qquad\text{vs.}\qquad \underbrace{(+1)(+1)(-1)}_{\text{columns}} = -1.

So $+1 = -1$ — impossible. No context-independent value assignment exists, and unlike KS this proof is state-independent: it holds for every quantum state, with no probabilities at all.

Operational consequences

Contextuality is not merely foundational. It is a certified resource: it powers advantages in certain communication and computation tasks, and "magic" — the non-stabilizer resource that makes universal quantum computation hard to simulate classically — is tied to contextuality on qubits. The same algebraic clash that refutes hidden variables fuels quantum computational power.

The takeaway

A noncontextual hidden-variable model assigns context-independent values respecting the algebra of commuting observables; the Kochen–Specker theorem and the Peres–Mermin square show no such model can match quantum mechanics, even with no entanglement or separation. Contextuality is the broad phenomenon of which Bell nonlocality is the spatially-separated special case.