On a gap in the proof of the generalised quantum Stein's lemma and its consequences for the reversibility of quantum resources
Mario Berta, Fernando G. S. L. Brandão, Gilad Gour, Ludovico Lami, Martin B. Plenio, Bartosz Regula, Marco Tomamichel
TL;DR
The paper identifies a gap in Brandao2010's generalized quantum Stein's lemma, showing that the claimed achievability part is not proven and that several foundational reversibility results for quantum resources under ARNG are unsettled. It analyzes the gap in detail, provides a counterexample via a varentropy-like construction, and discusses how reversibility might still be recovered under alternative Stein-type lemmas in restricted settings. The authors outline several promising approaches, including coherence-based and pseudo-entanglement frameworks, that can yield variant reversible results albeit with stronger restrictions or weaker guarantees than the original claims. They also clarify which related results remain valid or can be salvaged through independent proofs, and they emphasize that the central open problem—whether the generalised Stein's lemma is true in full generality—remains pressing for the broader program of reversible quantum resource theories.
Abstract
We show that the proof of the generalised quantum Stein's lemma [Brandão & Plenio, Commun. Math. Phys. 295, 791 (2010)] is not correct due to a gap in the argument leading to Lemma III.9. Hence, the main achievability result of Brandão & Plenio is not known to hold. This puts into question a number of established results in the literature, in particular the reversibility of quantum entanglement [Brandão & Plenio, Commun. Math. Phys. 295, 829 (2010); Nat. Phys. 4, 873 (2008)] and of general quantum resources [Brandão & Gour, Phys. Rev. Lett. 115, 070503 (2015)] under asymptotically resource non-generating operations. We discuss potential ways to recover variants of the newly unsettled results using other approaches.