# This Quantum World/Game

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## A quantum game

Here are the rules:[^1]

-   Two teams play against each other: Andy, Bob, and Charles (the
    \"players\") 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 each player is asked the **X** question, or one player is
    asked the **X** question and the two other players are asked the
    **Y** question.
-   The players win if the product of their answers is −1 in case only
    **X** questions are asked, and if the product of their answers is +1
    in case **Y** questions are asked. Otherwise they lose.
-   The players are not allowed to communicate with each other once the
    questions are asked. Before that, they are permitted to work out a
    strategy.

Is there a failsafe strategy? Can they make sure that they will win?
Stop to ponder the question.

Let us try pre-agreed answers, which we will call **X~A~**, **X~B~**,
**X~C~** and **Y~A~**, **Y~B~**, **Y~C~**. The winning combinations
satisfy the following equations:

:   \
    `<math>`{=html}

X_AY_BY_C=1,\\quad Y_AX_BY_C=1,\\quad Y_AY_BX_C=1,\\quad
X_AX_BX_C=-1.`</math>`{=html}

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Consider the first three equations. The product of their right-hand
sides equals +1. The product of their left-hand sides equals
**X~A~X~B~X~C~**, implying that **X~A~X~B~X~C~ = 1**. (Remember that the
possible values are ±1.) But if **X~A~X~B~X~C~ = 1**, then the fourth
equation **X~A~X~B~X~C~ = −1** obviously cannot be satisfied.

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

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[^1]: Lev Vaidman, \"Variations on the theme of the
    Greenberger-Horne-Zeilinger proof,\" *Foundations of Physics* 29,
    pp. 615-30, 1999.
