This paper is inspired by the PQ penny flip game. It employs group-theoretic concepts to study the original game and also its possible extensions. We show that the PQ penny flip game can be associated with the dihedral group $D_{8}$. We prove that within $D_{8}$ there exist precisely two classes of winning strategies for Q. We establish that there are precisely two different sequences of states that can guaranteed Q's win with probability $1.0$. We also show that the game can be played in the all dihedral groups $D_{8 n}$, $n \geq 1$, with any significant change. We examine what happens when Q can draw his moves from the entire $U(2)$ and we conclude that again, there are exactly two classes of winning strategies for Q, each class containing now an infinite number of equivalent strategies, but all of them send the coin through the same sequence of states as before. Finally, we consider general extensions of the game with the quantum player having $U(2)$ at his disposal. We prove that for Q to surely win against Picard, he must make both the first and the last move.
翻译:本文由 PQ penny 翻转游戏启发。 它使用组理论概念来研究最初的游戏及其可能的扩展。 我们显示, PQ penny 翻转游戏可以与 disheral Group $D ⁇ 8$ 相联。 我们证明, 在$D ⁇ 8} 美元范围内, 确实有两种类型的Q赢取策略。 我们确定, 确实有两种不同的状态序列可以保证Q赢得概率为1.0美元。 我们还显示, 该游戏可以在所有dhedral Group $D ⁇ 8 n$, $n\geq 1$中进行, 只要有任何重大改变。 我们检查当 Q 能够从整个 $U(2)$中抽出他的动作时会发生什么情况。 我们再次得出结论, Q 的赢取策略完全分为两类, 每个类包含无限数量的等效策略, 但所有这些国家都会通过之前的州顺序发送硬币。 最后, 我们考虑与量子玩家一起玩游戏的通用扩展范围, $U(2)$@ $\ g$ 1$ 1 。 我们证明, Q 要赢得Pical, 他必须首先和Pical 。