Quantum Pseudo-telepathy - The Mermin-Peres Magic Square Game

The Mermin-Peres Magic Square Game

An example of quantum pseudo-telepathy can be observed in the following two-player coordination game in which, in each round, one participant fills one row and the other fills one column of a 3x3 table with plus and minus signs.

The two players Alice and Bob are separated so that no communication between them is possible. In each round of the game Alice is told which row is selected for her to fill in, and Bob is told which column is selected for him. Alice is not told which column Bob must fill in, and Bob is not told which row Alice must fill in. Alice and Bob must both place the same sign in the cell shared by their row and column. Furthermore (and this is the catch), Alice has to fill the remainder of the row such that there is an even number of minus signs in that row, whilst Bob has to fill the remainder of the column such that there is an odd number of minus signs in that column.

It is easy to see that any prior agreement between Alice and Bob on the use of specific tables filled with + and – signs is not going to help them. The reason being that such tables simply do not exist: as these would be self-contradictory with the sum of the minus signs in the table being even based on row sums, and being odd when using column sums.

So, how can Alice and Bob succeed in their task?

The trick is for Alice and Bob to share an entangled quantum state and to use specific measurements on their components of the entangled state to derive the table entries. A suitable correlated state consists of a pair of Bell states:

$\left|\phi\right\rang = \frac{1}{\sqrt{2}} \bigg(\left|+\right\rang_a \otimes \left|+\right\rang_b + \left|-\right\rang_a \otimes \left|-\right\rang_b \bigg) \otimes \frac{1}{\sqrt{2}} \bigg(\left|+\right\rang_a \otimes \left|+\right\rang_b + \left|-\right\rang_a \otimes \left|-\right\rang_b \bigg)$

here |+> and |-> are eigenstates of the Pauli operator S_z with eigenvalues +1 and −1, respectively, whilst the subscripts a and b denote which component of each Bell state is going to Alice and which one goes to Bob.

Observables for these components can be written as products of the Pauli spin matrices:

$S_x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} , S_y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} , S_z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

Products of these Pauli spin operators can be used to fill the 3x3 table such that each row and each column contains a mutually commuting set of observables with eigenvalues +1 and −1, and with the product of the obervables in each row being the identity operator, and the product of observables in each column equating to minus the identity operator. This so-called Mermin-Peres magic square is shown in below table.

Effectively, while it is not possible to construct a 3x3 table with entries +1 and −1 such that the product of the elements in each row equals +1 and the product of elements in each column equals −1, it is possible to do so with the richer algebraic structure based on spin matrices.

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