Quantum pseudo-telepathy
The prefix pseudo refers to the fact that quantum pseudo-telepathy does not involve the exchange of information between any parties.
Instead, quantum pseudo-telepathy removes the need for parties to exchange information in some circumstances.
However, quantum pseudo-telepathy is a real-world phenomenon which can be verified experimentally.
It is thus an especially striking example of an experimental confirmation of Bell inequality violations.
A simple magic square game demonstrating nonclassical correlations was introduced by P. K. Aravind[3] based on a series of papers by N. David Mermin[4][5] and Asher Peres[6] and Adán Cabello[7][8] that developed simplifying demonstrations of Bell's theorem.
[9] This is a cooperative game featuring two players, Alice and Bob, and a referee.
The referee asks Alice to fill in one row, and Bob one column, of a 3×3 table with plus and minus signs.
It is easy to see that if Alice and Bob can come up with a classical strategy where they always win, they can represent it as a 3×3 table encoding their answers.
With a bit further analysis one can see that the best possible classical strategy can be represented by a table where each cell now contains both Alice and Bob's answers, that may differ.
It is possible to make their answers equal in 8 out of 9 cells, while respecting the parity of Alice's rows and Bob's columns.
This implies that if the referee asks for a row and column whose intersection is one of the cells where their answers match they win, and otherwise they lose.
Under the usual assumption that the referee asks for them uniformly at random, the best classical winning probability is 8/9.
This requires Alice and Bob to possess two pairs of particles with entangled states.
When Alice and Bob learn which column and row they must fill, each uses that information to select which measurements they should make to their particles.
The result of the measurements will appear to each of them to be random (and the observed partial probability distribution of either particle will be independent of the measurement performed by the other party), so no real "communication" takes place.
Note that Alice and Bob could be light years apart from one another, and the entangled particles will still enable them to coordinate their actions sufficiently well to win the game with certainty.
Playing N rounds requires that N entangled states (2N independent Bell pairs, see below) be shared in advance.
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.
are eigenstates of the Pauli operator Sx with eigenvalues +1 and −1, respectively, whilst the subscripts a, b, c, and d identify the components of each Bell state, with a and c going to Alice, and b and d going to Bob.
Observables for these components can be written as products of the Pauli matrices: Products of these Pauli spin operators can be used to fill the 3×3 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 observables in each row being the identity operator, and the product of observables in each column equating to minus the identity operator.
Effectively, while it is not possible to construct a 3×3 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.
It is possible to do that because all observables in a given row or column commute, so there exists a basis in which they can be measured simultaneously.
As long as the table above is used, the measurement results are guaranteed to always multiply out to +1 for Alice along her row, and −1 for Bob down his column.
Of course, each completely new round requires a new entangled state, as different rows and columns are not compatible with each other.
[14] In July 2022 a study reported the experimental demonstration of quantum pseudotelepathy via playing the nonlocal version of Mermin-Peres magic square game.
[18] In the game there are three players, Alice, Bob, and Carol playing against a referee.
Therefore, when the game is played the three questions of the referee x, y, z are drawn from the 4 options
Based on the question bit received, Alice, Bob, and Carol each respond with an answer a, b, c, also in the form of 0 or 1.
The players can formulate a strategy together prior to the start of the game.
When Alice, Bob, and Carol decide to adopt a quantum strategy they share a tripartite entangled state
