http://arxiv.org/abs/1505.04661
http://arxiv.org/abs/1507.00303

(I think the authors of the last one have now even a slightly simpler proof compared to what they had before and are planning a paper update.) In the first paper, you need some complex analysis to discuss the Hadamard three-line theorem. You then define a Renyi information quantity and apply the three-line theorem to it. In the second paper, you use properties of the “pinching map”, operator concavity of logarithm, and some other tricks. These are both tools that in my opinion are core quantum information tools. In view of the fact that the proof of strong subadditivity is already somewhat difficult, the above proofs are of comparable difficulty, and the proof of SSA is often presented in a QIT/QST course, it seems worthwhile to just go through the new results. There is more conceptual content as well, but it is interesting. For example, the recovery map discussed in the above papers is related to the Petz recovery map (AKA transpose channel), and you could connect to how you consider this to be a quantum Bayes theorem (discussed in Section IV-C of your paper http://arxiv.org/abs/1107.5849). The mileage gained out of the new results is big I think: you get finer statements about the Holevo bound (near saturation means approximately commuting states in a specific sense), faithfulness of squashed entanglement (a nice cloning argument related to the “bear attack” discussion above, presented in http://arxiv.org/abs/1410.4184), when discord is near to zero, etc. As far as quantum Shannon theory is concerned, the new results don’t change the capacity theorems or their proofs in any meaningful way (the converse proofs would just go the same — there is no apparent need for the extra recoverability term to establish a capacity theorem). Maybe if you’re looking at finer characterizations of capacity beyond the original setting, then these new results could play a role, but that is not really known (but right now I suspect it is not the case). For me, I am teaching QST this semester at LSU and am planning to incorporate the new results. We’ll see how it goes 🙂 As far as textbooks are concerned, I do think the new results have a place there, but I am personally biased on this one, having been involved in the new developments. But here consider that Petz discusses the equality cases of SSA and mono. of relative entropy at some length in his book “Quantum Information Theory and Quantum Statistics”, and since the new results are stronger than these, in a hypothetical scenario, perhaps Petz would include them in a book of his own (here I am speculating).