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The Product state page needs to be expanded. Sources need to be added. I'm sure QCQI mentions it, except it's not in the index, so I'll have to search the textbook.

It seems there are references on pages xx, 94, 106, 110 (x2), 202, 222, 358 (x3), 388, 398, 554 (x5), 555, 557, 558, 560 (x2), 561, 603 (x2), 605 (x4). I'll work over the next few days to describe each.

Nielsen and Chuang, p. xx: "[It was recently disproved that] the communication capacity of a quantum channel with product states is equal to the unconstrained capacity (i.e., the capacity with any entangled state allowed as input)". (More on this on p. 554 and on p. 605.)
Nielsen and Chuang, p. 605 (3/4): " King and Ruskai have made some promising headway on hte problem of showing that the product state capacity is the same as the capacity unrestricted to product states, but the general problem remains open."

Here is the Wikipedia search of all pages containing the text "product state" as well as the word quantum: [1]

A related term seems to be the "product state capacity," mentioned on p. 554 of Nielsen and Chuang. The Holevo–Schumacher–Westmoreland (HSW) theorem, in an analogous way to Shannon's noisy channel coding theorem, allows an "effective means for computing the product state capacity for a specified noisy channel ${\mathcal {E}}$ , and in some instances may even allow the derivation of an exact expression."