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

doi:10.22331/q-2023-09-07-1103

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

Mario Berta^1,2, Fernando G. S. L. Brandão^3,4, Gilad Gour⁵, Ludovico Lami^6,7,8,9, Martin B. Plenio⁶, Bartosz Regula^10,11, and Marco Tomamichel^12,13

¹Institute for Quantum Information, RWTH Aachen University, Aachen, Germany
²Department of Computing, Imperial College London, London, UK
³Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA
⁴AWS Center for Quantum Computing, Pasadena, CA, USA
⁵Department of Mathematics and Statistics, Institute for Quantum Science and Technology, University of Calgary, AB, Canada T2N 1N4
⁶Institut für Theoretische Physik und IQST, Universität Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany
⁷QuSoft, Science Park 123, 1098 XG Amsterdam, The Netherlands
⁸Korteweg–de Vries Institute for Mathematics, University of Amsterdam, Science Park 105-107, 1098 XG Amsterdam, The Netherlands
⁹Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
¹⁰Mathematical Quantum Information RIKEN Hakubi Research Team, RIKEN Cluster for Pioneering Research (CPR) and RIKEN Center for Quantum Computing (RQC), Wako, Saitama 351-0198, Japan
¹¹Department of Physics, Graduate School of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
¹²Center for Quantum Technologies, National University of Singapore, Singapore
¹³Department of Electrical and Computer Engineering, College of Design and Engineering, National University of Singapore, Singapore

Published:	2023-09-07, volume 7, page 1103
Eprint:	arXiv:2205.02813v4
Doi:	https://doi.org/10.22331/q-2023-09-07-1103
Citation:	Quantum 7, 1103 (2023).

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Updated version: The authors have uploaded version v5 of this work to the arXiv which may contain updates or corrections not contained in the published version v4. The authors left the following comment on the arXiv:

29 pages; in v2 we added Section V.D and Section VI, and corrected several small typos; v5 contains minor corrections in the discussion in Section V

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.

Featured image: A mathematical formulation of the generalised quantum Stein's lemma conjecture and its interpretation in terms of a quantum hypothesis testing task.

Popular summary

How efficiently can one detect the resources present in a quantum system? Qualitatively, the more resourceful a system is, the easier it is to detect its resources. To give a rigorous interpretation to this qualitative intuition, one needs to compute a quantity known as the Stein exponent of an associated quantum hypothesis testing (state discrimination) task. The generalised quantum Stein’s lemma, proposed by Brandão and Plenio in 2010, was a key piece of mathematical machinery that was designed to do precisely that. What is more, the resulting Stein exponent turned out to be given by a known resource measure, the regularised relative entropy. This makes intuitive sense, as a larger Stein exponent enhances the effectiveness of distinguishing a given state from all resourceless ones.

The applications of this result were manifold, but perhaps the most important was the identification of a framework for quantum resource manipulation that would become completely reversible. In this framework, discovered by Brandão and Plenio in 2008 and later generalised to almost all quantum resources by Brandão and Gour, the given resource could be freely converted from one form into the other at no (theoretical) loss. This mimics the classical thermodynamical behaviour of work and heat, which can be reversibly transformed into each other by Carnot cycles. For the case of entanglement theory, even more connections were established: notably, in this framework the distillable entanglement of any state is precisely equal to its Stein exponent.

In our work, we however report the existence of a serious gap in the original proof of the generalised quantum Stein’s lemma. Consequently, it remains uncertain whether this result is ultimately valid or not — the proof is incomplete, but we do not know of any counterexample to the original claim either. The results by Brandão and Plenio on reversibility of entanglement, and the subsequent ones on reversibility of general quantum resources, are now to be considered unproven. We discuss this state of affairs in detail, listing the affected results and explaining how to recover some of them. We also examine various ways of proving alternative but weaker forms of the generalised quantum Stein’s lemma. One of the goals of this paper is to stimulate further research on this problem, which appears to be one of the major open problems in the field of quantum entanglement theory and quantum resource theories in general, and whose complete solution would represent major progress in our understanding of the mysterious quantum world.

► BibTeX data

@article{Berta2023gapinproofof,
  doi = {10.22331/q-2023-09-07-1103},
  url = {https://doi.org/10.22331/q-2023-09-07-1103},
  title = {On a gap in the proof of the generalised quantum {S}tein's lemma and its consequences for the reversibility of quantum resources},
  author = {Berta, Mario and Brand{\~{a}}o, Fernando G. S. L. and Gour, Gilad and Lami, Ludovico and Plenio, Martin B. and Regula, Bartosz and Tomamichel, Marco},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1103},
  month = sep,
  year = {2023}
}

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