  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.
