The Peres-Mermin square, also called the Mermin-Peres magic square, is a 
arrangement of two-qubit
observables in which each row and column is a commuting context. Five context products
are the identity matrix 
and the remaining product is 
, giving a state-independent parity proof of the Kochen-Specker
theorem (Peres 1990, Mermin 1990, 1993).
Writing 
,
,
and 
for the Pauli matrices 
, 
, and 
, respectively, and writing juxtaposition for their Kronecker product, one form of the square is
![[XI IX XX; IY YI YY; XY YX ZZ].](/images/equations/Peres-MerminSquare/NumberedEquation1.svg) |
The product along every row and along the first two columns is 
, while the product down the third column is 
. A noncontextual assignment of values in 
would make the product of all six context equations equal
1 because every entry occurs twice. The specified context signs instead have product
,
which is a contradiction.
The anticommutation graph of these nine observables is the 
-rook graph
 |
where 
denotes the line graph. The associated hypergram
has six contexts, contextuality degree 1, and tolerated error per context 
(Muller and Giorgetti 2025, Muller and Saniga 2026).
