The maximal correlation coefficient measures the linear correlation in a bipartite distribution and contraction coefficients measure how much information is lost under a noisy channel. Remarkably, Raginsky established a close relation between these two concepts by showing that the $χ^2$ contraction coefficient equals the maximal correlation coefficient of the joint input/output distribution of the channel. In quantum theory, several generalizations of these concepts have been proposed, but none recover all the classical properties. Here we construct a framework in which the classical theory extends to the quantum setting. We introduce families of quantum maximal correlation coefficients and show that many impose limits on converting quantum states under local operations. We establish a family of quantum contraction coefficients are efficiently computable, yielding a generic efficient algorithm for mixing times of quantum channels with a full rank fixed point. Furthermore, we establish a quantum analogue of Raginsky's classical correspondence that relates these two families of quantities. To do this, we develop the operator-theoretic approach to Petz's family of non-commutative $L^{2}(p)$ spaces that extend the data processing inequality for variance to quantum theory.

翻译：最大相关系数衡量二分分布中的线性相关性，而收缩系数则度量噪声信道导致的信息损失程度。值得关注的是，Raginsky 通过证明卡方收缩系数等于信道联合输入/输出分布的最大相关系数，揭示了这两个概念间的紧密联系。在量子理论中，虽然已提出这些概念的多种推广形式，但均未能完全复现经典性质。本文构建了经典理论向量子领域扩展的统一框架：我们引入量子最大相关系数族，并证明其对局域操作下量子态转换的限制作用；同时建立可高效计算的量子收缩系数族，由此推导出关于全秩不动点量子信道混合时间的通用高效算法。此外，我们建立了联系这两族量的 Raginsky 经典对应关系的量子版本。为实现该目标，我们基于算子理论发展了 Petz 非交换 L²(p) 空间族的方法，将方差的数据处理不等式推广至量子领域。