School of Electrical and Electronic Engineering, University of Adelaide, Adelaide, South Australia, Australia.
PLoS One. 2012;7(5):e36404. doi: 10.1371/journal.pone.0036404. Epub 2012 May 11.
The N-player quantum games are analyzed that use an Einstein-Podolsky-Rosen (EPR) experiment, as the underlying physical setup. In this setup, a player's strategies are not unitary transformations as in alternate quantum game-theoretic frameworks, but a classical choice between two directions along which spin or polarization measurements are made. The players' strategies thus remain identical to their strategies in the mixed-strategy version of the classical game. In the EPR setting the quantum game reduces itself to the corresponding classical game when the shared quantum state reaches zero entanglement. We find the relations for the probability distribution for N-qubit GHZ and W-type states, subject to general measurement directions, from which the expressions for the players' payoffs and mixed Nash equilibrium are determined. Players' N x N payoff matrices are then defined using linear functions so that common two-player games can be easily extended to the N-player case and permit analytic expressions for the Nash equilibrium. As a specific example, we solve the Prisoners' Dilemma game for general N ≥ 2. We find a new property for the game that for an even number of players the payoffs at the Nash equilibrium are equal, whereas for an odd number of players the cooperating players receive higher payoffs. By dispensing with the standard unitary transformations on state vectors in Hilbert space and using instead rotors and multivectors, based on Clifford's geometric algebra (GA), it is shown how the N-player case becomes tractable. The new mathematical approach presented here has wide implications in the areas of quantum information and quantum complexity, as it opens up a powerful way to tractably analyze N-partite qubit interactions.
我们分析了使用爱因斯坦-波多尔斯基-罗森（Einstein-Podolsky-Rosen，EPR）实验作为基础物理设置的 N 人量子博弈。在这种设置中，玩家的策略不是交替量子博弈论框架中的幺正变换，而是在自旋或极化测量的两个方向之间进行经典选择。因此，玩家的策略与他们在经典游戏的混合策略版本中的策略相同。在 EPR 设置中，当共享量子态达到零纠缠时，量子博弈就简化为相应的经典博弈。我们找到在一般测量方向下，N 量子 GHZ 和 W 态的概率分布的关系，从而确定玩家的收益和混合纳什均衡的表达式。然后，我们使用线性函数定义玩家的 N x N 收益矩阵，以便可以轻松地将常见的两人游戏扩展到 N 人游戏，并允许为纳什均衡提供解析表达式。作为一个具体的例子，我们为一般的 N ≥ 2 解决了囚徒困境游戏。我们发现该游戏的一个新特性，即对于偶数个玩家，纳什均衡的收益相等，而对于奇数个玩家，合作玩家的收益更高。通过放弃希尔伯特空间中状态向量的标准幺正变换，而代之以基于 Clifford 几何代数（geometric algebra，GA）的转子和多向量，我们展示了如何使 N 人案例变得易于处理。这里提出的新数学方法在量子信息和量子复杂性领域具有广泛的意义，因为它为可分析地分析 N 部分量子比特相互作用提供了一种强大的方法。
