In classical information theory, the maximal correlation and $χ^{2}$-contraction coefficient establish limits on distributed and sequential processing. Two distinct quantum maximal correlation coefficients have been proposed, but they do not extend all the classical results. Building on work of Petz, we use the family of non-commutative $L^{2}(p)$ spaces that extend the data processing inequality for variance to quantum theory to extend the classical results to quantum theory. We introduce families of quantum maximal correlation coefficients and identify quantum $χ^{2}$-divergences as non-commutative generalizations of the variance of the likelihood ratio. We establish a family of maximal correlation coefficients that must all be ordered on a single copy level for an arbitrary number of copies of one state to be able to be converted to a single copy of another target state under local operations. We prove the equivalent characterizations of perfect classical correlation extraction via local operations in quantum theory. We clarify the relationship between maximal correlation and $χ^{2}$-contraction coefficients by proving they are the same operator norms evaluated on distinct maps. Then we establish new equivalent conditions to the saturation of the data processing inequality for $χ^{2}$-divergences. This implies previous saturation results for the $χ^{2}$ and sandwiched Rényi divergences. Finally, we establish the quantum maximal correlation coefficients and $χ^{2}$-contraction coefficients are often efficiently computable. This results in a generic method for efficiently computing mixing times of time-homogeneous quantum Markov chains with a unique full rank fixed point.

翻译：在经典信息论中，最大相关系数和$χ^{2}$-收缩系数为分布式与顺序处理建立了极限。已有两种不同的量子最大相关系数被提出，但它们并未完全推广所有经典结果。基于Petz的工作，我们利用非交换$L^{2}(p)$空间族——该空间将方差的经典数据处理不等式推广至量子理论——从而将经典结果拓展到量子理论中。我们引入了量子最大相关系数族，并将量子$χ^{2}$-散度识别为似然比方差的非交换推广。我们建立了一族最大相关系数，其必须在单副本层面全部有序，才能使得任意数量副本的某个态通过局域操作转化为目标态的单副本。我们证明了量子理论中通过局域操作实现完美经典关联提取的等价刻画。通过证明最大相关系数与$χ^{2}$-收缩系数是同一算子范数在不同映射上的取值，我们阐明了二者之间的关系。随后，我们建立了$χ^{2}$-散度的数据处理不等式饱和的新等价条件，这蕴含了先前关于$χ^{2}$散度与夹层Rényi散度的饱和结果。最后，我们证明了量子最大相关系数与$χ^{2}$-收缩系数通常可高效计算，从而为计算具有唯一满秩不动点的时齐量子马尔可夫链的混合时间提供了一种通用高效方法。