In this article, we investigate relative position between a pair of spin model subfactors of the hyperfinite type $II_1$ factor $R$ arising from two complex Hadamard matrices of order $2$ as well as order $4$. More precisely, we characterize when the two subfactors are equal, compute the Pimsner-Popa probabilistic constant and the Connes-St{\o}rmer relative entropy between them. To the best of our knowledge, this article is the first instance in the literature that the exact value of the Connes-St{\o}rmer relative entropy for a pair of (non-trivial) subfactors has been obtained. We construct en route a family of potentially new subfactors of $R$. All these subfactors are irreducible with Jones index $4n,n\in\mathbb{N}$. As a corollary, a rigidity of the angle between the two subfactors is established. Finally, as a pleasant application of the relative entropy, we characterize when the pair of spin model subfactors form a commuting square.

翻译：本文研究超有限型$II_1$因子$R$中由两个阶数分别为2和4的复Hadamard矩阵导出的一对自旋模型子因子之间的相对位置。具体而言，我们刻画了这两个子因子相等的条件，计算了它们之间的Pimsner-Popa概率常数和Connes-Størmer相对熵。据我们所知，本文是文献中首次获得一对（非平凡）子因子Connes-Størmer相对熵精确值的研究。在过程中，我们构造了一族潜在的$R$的新子因子。所有这些子因子均为不可约的，且Jones指标为$4n,n\in\mathbb{N}$。作为推论，我们建立了两个子因子之间角度的刚性。最后，作为相对熵的一个有趣应用，我们刻画了这对自旋模型子因子构成交换方的条件。