Past exam of the computer science course of the University of Oxford / 2019 / Quantum information / 2 / c

For deterministic classical responses, denote the two possible local answers by $x_0,x_1$, $y_0,y_1$ and $z_0,z_1$. Certainty in the GHZ game would requireTaking the exclusive or of all four equations gives zero on the left, since every answer occurs twice, but one on the right. This contradiction rules out deterministic certainty. Pre-shared randomness cannot cure it: conditioning on the random seed fixes a deterministic response table, and none succeeds on every allowed input. Thus noncommunicating classical players cannot win with certainty. No uniform distribution on the referee's questions is needed for this worst-case claim.

For the quantum strategy, share the GHZ state and measure the Pauli X gate observable when the local question is 0, or the Pauli Y gate observable when it is 1. Return bit 0 for measurement eigenvalue $+1$ and bit 1 for eigenvalue $-1$. The four relevant operator identities areFor the first identity, the three bit flips exchange $|000⟩$ and $|111⟩$. For each remaining identity, the two Y actions contribute $i^2=-1$ on $|000⟩$ and $(-i{)}^2=-1$ on $|111⟩$, so they exchange the terms with a minus sign. Local measurements on separate systems commute, and the product of their observed eigenvalues is therefore fixed by the displayed joint eigenvalue. That product equals $(-1{)}^{x+y+z}$, giving exactly the required output parity for each promised input. Hence the quantum strategy wins with certainty without communication. This is the GHZ theorem in nonlocal-game form; it does not let a player control another player's individual outcome.