6 A quantum game

Let’s play a game. We form two teams of three, the “players” (Andy, Bob, and Charles) and the “inter­roga­tors”.[1] The rules of this game are as follows:

  • Either all players are asked for the value of X, or one player is asked for the value of X while the two other players are asked for the value of Y.
  • The pos­sible values of both X and Y are +1 and −1.
  • If all players are asked for the value of X, they win if (and only if) the product of their answers equals −1. Oth­er­wise they win if (and only if) the product of their answers equals +1.

Once the ques­tions are asked, the players are no longer allowed to com­mu­ni­cate with each other. Prior to that, they may work out a strategy. Is there a fail-​​safe strategy? Can they make sure that they will win? Ponder this before you proceed.

The obvious strategy is to use pre-​​agreed answers.

Let’s call them XA, XB, XC, and YA, YB, YC.

Now try this: Assign values (+1 or −1) to the fol­lowing vari­ables in such a way that the product of the three X values equals −1 while the product of the Y values in two of the three columns equals the X value in the remaining column — or else explain why this can’t be done.
XA   XB   XC
YA   YB   YC

Here is why it can’t be done. The win­ning com­bi­na­tions sat­isfy the fol­lowing equa­tions:
XA   XB   XC = −1
XA   YB   YC = +1
YA   XB   YC = +1
YA   YB   XC = +1

Because the squares of the Y’s are equal to 1, the product of the left-​​hand sides of the last three equa­tions is

XA XB XC (YA)2 (YB)2 (YC)2 = XA XB XC,

while the product of their right-​​hand sides is +1. Obvi­ously these three equa­tions cannot be sat­is­fied as long as the first equa­tion holds. Upshot: pre-​​agreed answers offer no fail-​​safe strategy.

And yet there is such a strategy.

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1. [↑] Vaidman, L. (1999). Vari­a­tions on the theme of the Greenberger-​​Horne-​​Zeilinger proof, Foun­da­tions of Physics 29, 615–630.