Legón Alexis R, Medina Ernesto
Departamento de Física, Universidad Técnica Federico Santa María, Av. España 1680, Casilla 110 V, Valparaíso 2390123, Chile.
Laboratorio de Física Estadística de Medios Desordenados, Instituto Venezolano de Investigaciones Científicas (IVIC) Carretera Panamericana, Km 11, Altos de Pipe, Caracas 1020A, Venezuela.
Entropy (Basel). 2023 Aug 16;25(8):1222. doi: 10.3390/e25081222.
A joint probability formalism for quantum games with noise is proposed, inspired by the formalism of non-factorizable probabilities that connects the joint probabilities to quantum games with noise. Using this connection, we show that the joint probabilities are non-factorizable; thus, noise does not generically destroy entanglement. This formalism was applied to the Prisoner's Dilemma, the Chicken Game, and the Battle of the Sexes, where noise is coupled through a single parameter μ. We find that for all the games except for the Battle of the Sexes, the Nash inequalities are maintained up to a threshold value of the noise. Beyond the threshold value, the inequalities no longer hold for quantum and classical strategies. For the Battle of the sexes, the Nash inequalities always hold, no matter the noise strength. This is due to the symmetry and anti-symmetry of the parameters that determine the joint probabilities for that game. Finally, we propose a new correlation measure for the games with classical and quantum strategies, where we obtain that the incorporation of noise, when we have quantum strategies, does not affect entanglement, but classical strategies result in behavior that approximates quantum games with quantum strategies without the need to saturate the system with the maximum value of noise. In this manner, these correlations can be understood as entanglement for our game approach.
受将联合概率与含噪声量子博弈联系起来的不可分解概率形式主义的启发，我们提出了一种用于含噪声量子博弈的联合概率形式主义。利用这种联系，我们表明联合概率是不可分解的；因此，噪声通常不会破坏纠缠。这种形式主义被应用于囚徒困境、斗鸡博弈和性别大战，其中噪声通过单个参数μ耦合。我们发现，除了性别大战之外，对于所有博弈，纳什不等式在噪声的阈值以下都成立。超过阈值后，对于量子和经典策略，不等式不再成立。对于性别大战，无论噪声强度如何，纳什不等式总是成立。这是由于决定该博弈联合概率的参数的对称性和反对称性。最后，我们为具有经典和量子策略的博弈提出了一种新的关联度量，我们发现，当存在量子策略时，噪声的加入不会影响纠缠，但经典策略会导致一种行为，这种行为在不需要用最大噪声值使系统饱和的情况下近似于具有量子策略的量子博弈。通过这种方式，这些关联可以被理解为我们博弈方法中的纠缠。
