Quantum contextuality

Remarks

Example: the Peres–Mermin square on $\C^4 = \C^2 \otimes \C^2$ consists of nine products of Pauli matrices arranged in a $3 \times 3$ grid. Its rows are $X \otimes I, I \otimes X, X \otimes X$, then $I \otimes Z, Z \otimes I, Z \otimes Z$, then $X \otimes Z, Z \otimes X, Y \otimes Y$. The entries of each row and of each column commute, and every entry squares to the identity. Each row multiplies to $I$, and the columns multiply to $I$, $I$ and $-I$. A value assignment would give each entry a value $\pm 1$ whose products are $+1$ on each row and on the first two columns, but $-1$ on the last column. The product of all nine values would then be $+1$ by rows and $-1$ by columns, so the set is contextual. Non-example: any set of pairwise commuting observables is noncontextual, since a common eigenvector $\psi$ gives the assignment $A \mapsto$ eigenvalue of $A$ on $\psi$.