I'm sensing a lot of buzz about potential re-proofs of the Generalized Quantum Stein's Lemma - a generalization of the quantum counterpart to the classical Stein's Lemma, which is of some importance in statistical inference and hypothesis testing.
Apparently, Brandão and Plenio provided an original proof of the lemma around 2008, which was relied upon in many other subsequent results, but in 2023 an error in that proof was identified by Berta, Brandão, Gour, Lami, Plenio, Regula, and Tomamichel.
Nonetheless, some recent (August 2024) preprints - a first by Hayashi and Yamasaki and a second by Lami (one of the authors of the '23 paper) - claim to reestablish the Generalized Quantum Stein's Lemma, and if correct help put those subsequent results back on more solid ground.
Regardless of the merits of the reproofs, though, it's hard for me to find a simple explanation as to even the significance of the statement of the lemma. Thus, what's a TL/DR statement of the lemma and of its importance? To make it concrete, are there any presently known specific testable consequences of the lemma - either for quantum computing or for quantum communication?
For example, does the lemma indicate that local operation and classical communication (LOCC) resources have a graceful growth rate for entanglement distillation? Is it a "good" lemma for quantum computing and/or quantum information, such as implying that resources for (e.g.) quantum state discrimination don't grow too quickly?
