Example
This phenomenon can be shown with a pentagram, which is a graph with 5 vertices and 5 edges. Each vertex can be colored either red or blue. An edge is said to match if both of its vertices have the same color. In a hidden variable model, the total number of mismatches over all of the edges has to be an even number due to cyclicity, i.e. 0, 2 or 4. Therefore, with a probability mixture over hidden variable assignments, the expectation value of the sum of mismatches over all of the 5 edges lies between 0 and 4.
A large number of KCBS pentagrams can be imagined, each with colorings hidden. On each pentagram, a theoretical observer uncovers 2 vertices that share a common edge. Doing this shows that no matter which edge is chosen, it always ends with finding blue-blue with a probability of , red-blue with , and blue-red with . So, the expectation value of the sum of mismatches is .
To explain, each pentagram is a 3D quantum system with an orthonormal basis , and is initialized to . Each vertex is assigned a 1D projector projecting to , n = 0, ..., 4 .
Adjacent projectors commute. Projected vertices are colored red; otherwise, blue.
