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We show that the proof of the generalised quantum Stein's lemma [Brand\~ao &Plenio, Commun. Math. Phys. 295, 791 (2010)] is not correct due to a gap in theargument leading to Lemma III.9. Hence, the main achievability result ofBrand\~ao & Plenio is not known to hold. This puts into question a number ofestablished results in the literature, in particular the reversibility ofquantum entanglement [Brand\~ao & Plenio, Commun. Math. Phys. 295, 829 (2010);Nat. Phys. 4, 873 (2008)] and of general quantum resources [Brand\~ao & Gour,Phys. Rev. Lett. 115, 070503 (2015)] under asymptotically resourcenon-generating operations. We discuss potential ways to recover variants of thenewly unsettled results using other approaches.

On a gap in the proof of the generalised quantum Stein's lemma and its consequences for the reversibility of quantum resources