In classical information theory, the Doeblin coefficient of a classical channel provides an efficiently computable upper bound on the total-variation contraction coefficient of the channel, leading to what is known as a strong data-processing inequality. Here, we investigate quantum Doeblin coefficients as a generalization of the classical concept. In particular, we define various new quantum Doeblin coefficients, one of which has several desirable properties, including concatenation and multiplicativity, in addition to being efficiently computable. We also develop various interpretations of two of the quantum Doeblin coefficients, including representations as minimal singlet fractions, exclusion values, reverse max-mutual and oveloH informations, reverse robustnesses, and hypothesis testing reverse mutual and oveloH informations. Our interpretations of quantum Doeblin coefficients as either entanglement-assisted or unassisted exclusion values are particularly appealing, indicating that they are proportional to the best possible error probabilities one could achieve in state-exclusion tasks by making use of the channel. We also outline various applications of quantum Doeblin coefficients, ranging from limitations on quantum machine learning algorithms that use parameterized quantum circuits (noise-induced barren plateaus), on error mitigation protocols, on the sample complexity of noisy quantum hypothesis testing, and on mixing, distinguishability, and decoupling times of time-varying channels. All of these applications make use of the fact that quantum Doeblin coefficients appear in upper bounds on various trace-distance contraction coefficients of a channel. Furthermore, in all of these applications, our analysis using Doeblin coefficients provides improvements of various kinds over contributions from prior literature, both in terms of generality and being efficiently computable.

翻译：在经典信息论中，经典信道的Doeblin系数为该信道的全变差收缩系数提供了可高效计算的上界，进而引出了所谓强数据处理不等式。本文研究了经典概念的推广——量子Doeblin系数。具体而言，我们定义了多种新型量子Doeblin系数，其中一种兼具可加性、可乘性及高效可计算性等优良性质。我们还对其中两种量子Doeblin系数给出了多重诠释，包括其作为最小单重态分数、排除值、逆向最大互信息和oveloH信息、逆向鲁棒性以及假设检验逆向互信息和oveloH信息的表示形式。特别值得关注的是，我们将量子Doeblin系数诠释为纠缠辅助型或非辅助型排除值，表明该系数与利用该信道在态排除任务中可实现的最优错误概率成正比。本文还概述了量子Doeblin系数的多种应用场景，涵盖参数化量子电路（噪声诱导的贫瘠高原）对量子机器学习算法的限制、误差缓解协议、含噪量子假设检验的样本复杂度，以及时变信道的混合、可区分性和退耦合时间。所有应用均基于量子Doeblin系数作为信道各类迹距离收缩系数上界这一特性。与既有文献相比，本文基于Doeblin系数的分析在通用性与高效可计算性两个维度上均实现了改进。