Uniform continuity bounds on entropies are generally expressed in terms of a single distance measure between a pair of probability distributions or quantum states, typically, the total variation distance or trace distance. However, if an additional distance measure between the probability distributions or states is known, then the continuity bounds can be significantly strengthened. Here, we prove a tight uniform continuity bound for the Shannon entropy in terms of both the local- and total variation distances, sharpening an inequality proven in [I. Sason, IEEE Trans. Inf. Th., 59, 7118 (2013)]. We also obtain a uniform continuity bound for the von Neumann entropy in terms of both the operator norm- and trace distances. The bound is tight when the quotient of the trace distance by the operator norm distance is an integer. We then apply our results to compute upper bounds on the quantum- and private classical capacities of channels. We begin by refining the concept of approximate degradable channels, namely, $\varepsilon$-degradable channels, which are, by definition, $\varepsilon$-close in diamond norm to their complementary channel when composed with a degrading channel. To this end, we introduce the notion of $(\varepsilon,\nu)$-degradable channels; these are $\varepsilon$-degradable channels that are, in addition, $\nu$-close in completely bounded spectral norm to their complementary channel, when composed with the same degrading channel. This allows us to derive improved upper bounds to the quantum- and private classical capacities of such channels. Moreover, these bounds can be further improved by considering certain unstabilized versions of the above norms. We show that upper bounds on the latter can be efficiently expressed as semidefinite programs. We illustrate our results by obtaining a new upper bound on the quantum capacity of the qubit depolarizing channel.

翻译：熵的均匀连续性界通常以概率分布或量子态之间单一距离度量（典型如全变差距离或迹距离）来表述。然而，若已知概率分布或态之间的额外距离度量，则连续性界可被显著加强。本文针对香农熵证明了基于局部变差距离和全变差距离的紧致均匀连续性界，强化了[I. Sason, IEEE Trans. Inf. Th., 59, 7118 (2013)]中的不等式。此外，我们得到了冯·诺依曼熵在算子范数距离和迹距离下的一致连续性界，且当迹距离与算子范数距离的商为整数时该界是紧致的。进一步将上述结果应用于信道量子容量和私有经典容量的上界计算。首先，我们精炼了近似可退化信道的概念——即满足定义中与互补信道在金刚范数下ε-接近（通过退化信道复合）的ε-可退化信道。为此引入(ε,ν)-可退化信道的概念：此类信道不仅满足ε-可退化性，且与相同退化信道复合后，其完全有界谱范数与互补信道之差的ν-邻域内。基于此，我们推导出此类信道量子容量和私有经典容量的改进上界，并通过考虑上述范数的非稳定化版本进一步优化。研究表明，后者上界可高效表示为半定规划问题。最终以量子比特退极化信道的量子容量新上界为例验证结论有效性。