This Quantum World

Prelude to GHZ

The bare facts

Let's play a game. Here are the rules:

There are two opponent teams: the "players" (Andy, Bob, and Charles) versus the "interrogators."
Each player is asked either "What is the value of X" or "What is the value of Y?"
Only two answers are allowed: +1 or –1.
Either all players are asked the first question (about the value of X), or one player is asked the first question and two players are asked the second question (about the value of Y).
The players win if the product of their answers is –1 in case everyone is asked the first question, and if the product of their answers is +1 in the case that Y-questions are asked. Otherwise they loose.
The players are not allowed to communicate which each other once the questions have been asked. Before that, they are permitted to work out a strategy.

Is there a failsafe strategy? Can the players make sure that they will win?

RFA image

Let us try pre-agreed answers. Let's call them X_A, X_B, X_C and Y_A, Y_B, Y_C. The winning combinations satisfy the following equations:

X_A Y_B Y_C = +1
Y_A X_B Y_C = +1,
Y_A Y_B X_C = +1,
X_A X_B X_C = –1.

The product of the left-hand sides of the first three equations equals X_A X_B X_C. (Remember, the possible values are +1 and –1.) The product of the right-hand sides of these three equations equals +1, implying that X_A X_B X_C = +1. But if this holds, then the fourth equation obviously cannot hold.

The bottom line: there is no failsafe strategy with pre-agreed answers.

This game has been adapted from L. Vaidman, "Variations on the theme of the Greenberger-Horne-Zeilinger proof," Foundations of Physics 29, 615–30 (1999).

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