  The quantum capacity of a noisy quantum channel determines the maximal rate
at which we can code reliably over asymptotically many uses of the channel, and
it characterizes the channel's ultimate ability to transmit quantum information
coherently. In this paper, we derive single-letter upper bounds on the quantum
and private capacities of quantum channels. The quantum capacity of a quantum
channel is always no larger than the quantum capacity of its extended channels,
since the extensions of the channel can be considered as assistance from the
environment. By optimizing the parametrized extended channels with specific
structures such as the flag structure, we obtain new upper bounds on the
quantum capacity of the original quantum channel. Furthermore, we extend our
approach to estimating the fundamental limits of private communication and
one-way entanglement distillation. As notable applications, we establish
improved upper bounds to the quantum and private capacities for fundamental
quantum channels of interest in quantum information, some of which are also the
sources of noise in superconducting quantum computing. In particular, our upper
bounds on the quantum capacities of the depolarizing channel and the
generalized amplitude damping channel are strictly better than previously
best-known bounds for certain regimes.
