  The maximum rate at which classical information can be reliably transmitted
per use of a quantum channel strictly increases in general with $N$, the number
of channel outputs that are detected jointly by the quantum joint-detection
receiver (JDR). This phenomenon is known as superadditivity of the maximum
achievable information rate over a quantum channel. We study this phenomenon
for a pure-state classical-quantum (cq) channel and provide a lower bound on
$C_N/N$, the maximum information rate when the JDR is restricted to making
joint measurements over no more than $N$ quantum channel outputs, while
allowing arbitrary classical error correction. We also show the appearance of a
superadditivity phenomenon---of mathematical resemblance to the aforesaid
problem---in the channel capacity of a classical discrete memoryless channel
(DMC) when a concatenated coding scheme is employed, and the inner decoder is
forced to make hard decisions on $N$-length inner codewords. Using this
correspondence, we develop a unifying framework for the above two notions of
superadditivity, and show that for our lower bound to $C_N/N$ to be equal to a
given fraction of the asymptotic capacity $C$ of the respective channel, $N$
must be proportional to $V/C^2$, where $V$ is the respective channel dispersion
quantity.
