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 average error probability of sending M
messages through a c-q channel is upper bounded by the error of distinguishing
the joint state between channel input and output against (M−1)-many products
of its marginals. (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 that rely on the Hayashi-Nagaoka operator inequality.
In particular, we obtain the tightest achievable ϵ-one-shot capacity
for c-q channel heretofore, and it improves 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 bounds for data
compression with quantum side information, entanglement-assisted classical
communication over quantum channels, and various quantum network
information-processing protocols