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

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... 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.