The generalised quantum Stein's lemma, and reversibility under asymptotically non-entangling operations

Proved, with a source.

The Stein exponent for testing an entangled state against separable states equals the regularised relative entropy of entanglement. This gives reversibility when operations may generate vanishing entanglement. The original 2010 proof had a gap, found in 2022. Hayashi and Yamasaki, and Lami, then gave independent proofs in 2024, and the Hayashi–Yamasaki proof has been formalised in Lean.

Connections

Part of: Conversion
See also: Relative entropy of entanglement is additive
See also: Entanglement is reversible under all non-entangling operations

Sources

A reversible theory of entanglement and its relation to the second law. F. G. S. L. Brandão, M. B. Plenio, 2007.
On a gap in the proof of the generalised quantum Stein's lemma and its consequences for the reversibility of quantum resources. M. Berta, F. G. S. L. Brandão, G. Gour, L. Lami, M. B. Plenio, B. Regula, M. Tomamichel, 2022.
Generalized quantum Stein's lemma and second law of quantum resource theories. M. Hayashi, H. Yamasaki, 2024.
A solution of the generalised quantum Stein's lemma. L. Lami, 2024.
A formalization of the generalized quantum Stein's lemma in Lean. A. Meiburg, L. A. Lessa, R. R. Soldati, 2025.
Generalised quantum Stein's lemma more robust than ever. F. Girardi, K.-Y. Lee, M. Hayashi, L. Lami, 2026.

Checked 2026-10-30.