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：对熵的统一连续性界通常表示为概率分布或量子态对之间单个距离测度的函数，典型地包括全变差距离或迹距离。然而，若已知概率分布或量子态之间的额外距离测度，则可显著强化连续性界。本文证明了香农熵在局部全变差距离与全变差距离双重约束下的紧统一连续性界，强化了[I. Sason, IEEE Trans. Inf. Th., 59, 7118 (2013)]中的不等式。同时，我们获得了冯·诺依曼熵在算子范数距离与迹距离双重约束下的统一连续性界，当迹距离与算子范数距离之比为整数时该界达到紧致。进而将所得结果应用于计算信道量子容量与私有经典容量的上界：首先精化近似退化信道的概念，即ε-退化信道——定义为与其互补信道在金刚石范数下ε-接近的信道（通过退化信道合成）；为此引入(ε,ν)-退化信道的概念，即额外满足与其互补信道在完全有界谱范数下ν-接近的ε-退化信道（通过相同退化信道合成）。由此推导出此类信道量子容量与私有经典容量的改进上界，且通过考虑上述范数的某些非稳定化版本可进一步优化。研究表明后者上界可高效表示为半定规划。通过量子比特退极化信道量子容量新上界的计算验证了所得结果。