  Achievability in information theory refers to demonstrating a coding strategy
that accomplishes a prescribed performance benchmark for the underlying task.
In quantum information theory, the crafted Hayashi-Nagaoka operator inequality
is an essential technique in proving a wealth of one-shot achievability bounds
since it effectively resembles a union bound in various problems. In this work,
we show that the pretty-good measurement naturally plays a role as the union
bound as well. A judicious application of it considerably simplifies the
derivation of one-shot achievability for classical-quantum (c-q) channel coding
via an elegant three-line proof.
  The proposed analysis enjoys the following favorable features. (i) The
established one-shot bound admits a closed-form expression as in the celebrated
Holevo-Helstrom Theorem. Namely, the error probability of sending $M$ messages
through a c-q channel is upper bounded by the minimum error of distinguishing
the joint channel input-output state against $(M-1)$ decoupled products states.
(ii) Our bound directly yields asymptotic results in the large deviation, small
deviation, and moderate deviation regimes in a unified manner. (iii) The
coefficients incurred in applying the Hayashi-Nagaoka operator inequality are
no longer needed. Hence, the derived one-shot bound sharpens existing results
relying on the Hayashi-Nagaoka operator inequality. In particular, we obtain
the tightest achievable $\epsilon$-one-shot capacity for c-q channel coding
heretofore, improving the third-order coding rate in the asymptotic scenario.
(iv) Our result holds for infinite-dimensional Hilbert space. (v) The proposed
method applies to deriving one-shot achievability for classical data
compression with quantum side information, entanglement-assisted classical
communication over quantum channels, and various quantum network
information-processing protocols.
