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Right, they are quite similar. The Holevo bound is a bound on the amount of accessible information between your quantum system and your classical system. The I(X;B) object written in the HSW theorem wikipedia page is actually this bound, while the $\chi$ there is the Holevo rate, or product state capacity. What HSW showed was that if you took many copies of ...

Here's a relatively simple proof just based on the data processing inequality (DPI) for the relative entropy $D(\rho\|\sigma) = \mathrm{tr}[\rho (\log \rho - \log \sigma)]$ -- if you're willing to accept the DPI as a basis for a formal proof. Recall that the DPI says that for any channel $\Phi$ we have $$ D(\rho \|\sigma) \geq D(\Phi(\rho)\|\Phi(\sigma)). $$...

To expand on @Purva Tharke's comment, the strong subadditivity inequality states: $$H(ABC)+H(C) \le H(AC) + H(BC)$$ $$=H(ABC)+H(C) +H(C) -H(C) \le H(AC) + H(BC)$$ $$=H(AB|C) \le H(A|C) + H(B|C)$$ $$=0\le H(A|C) + H(B|C) - H(AB|C)=H(A;B|C)$$ Edit: A good proof of SS for entropies can be found in Nielsen and Chuang, in case you wanted to take a look.

Here's a guess: they might be related to entanglement-breaking channels (also known as measure-and-prepare channels, quantum-classical channels, etc.). Any channel of the form, $$ \Phi(\rho) = \sum\limits_{k} \operatorname{Tr}\left( M_{k} \rho \right) \sigma_{k} , \text{ where } M_{k}\geq0,\sum\limits_{k}^{} M_{k} = \mathbb{I}, $$ are POVM elements and $\{ \...