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Entanglement theory and the second law of thermodynamics

Nature Physics volume 4, pages 873–877 (2008)Cite this article

17 May 2022 Editor’s Note: Readers are alerted that the conclusions of this Article rely on a proof of the generalised quantum Stein's lemma in Commun. Math. Phys. 295, 791 (2010) that has been called into question. Details are available in a preprint https://arxiv.org/abs/2205.02813 . A further editorial response will follow the resolution of these issues for this Article.

Abstract

Entanglement is central both to the foundations of quantum theory and, as a novel resource, to quantum information science. The theory of entanglement establishes basic laws that govern its manipulation, in particular the non-increase of entanglement under local operations on the constituent particles. Such laws aim to draw from them formal analogies to the second law of thermodynamics; however, whereas in the second law the entropy uniquely determines whether a state is adiabatically accessible from another, the manipulation of entanglement under local operations exhibits a fundamental irreversibility, which prevents the existence of such an order. Here, we show that a reversible theory of entanglement and a rigorous relationship with thermodynamics may be established when considering all non-entangling transformations. The role of the entropy in the second law is taken by the asymptotic relative entropy of entanglement in the basic law of entanglement. We show the usefulness of this approach to general resource theories and to quantum information theory.

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Figure 1: Asymptotic entanglement conversion.
Figure 2: Relative entropy of entanglement and global robustness of entanglement.
Figure 3: Reversibility of entanglement manipulation.
Figure 4: Quantum version of Stein’s lemma.

Change history

  • 17 May 2022

    Editor’s Note: Readers are alerted that the conclusions of this Article rely on a proof of the generalised quantum Stein's lemma in Commun. Math. Phys. 295, 791 (2010) that has been called into question. Details are available in a preprint https://arxiv.org/abs/2205.02813. A further editorial response will follow the resolution of these issues for this Article.

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Acknowledgements

We gratefully thank K. Audenaert, J. Eisert, A. Grudka, M. Horodecki, R. Horodecki, S. Virmani and R.F. Werner for useful discussions and correspondence. This work is part of the QIP-IRC supported by EPSRC and the Integrated Project Qubit Applications (QAP) supported by the IST directorate and was supported by the Brazilian agency Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and a Royal Society Society Wolfson Research Merit Award.

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Authors and Affiliations

  1. QOLS, Blackett Lab, Imperial College London, Prince Consort Road, London SW7 2BW, UK

    Fernando G. S. L. Brandão & Martin B. Plenio

  2. Institute for Mathematical Sciences, Imperial College London, 53 Exhibition Road, London SW7 2PG, UK

    Fernando G. S. L. Brandão & Martin B. Plenio

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  1. Fernando G. S. L. Brandão
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Correspondence to Fernando G. S. L. Brandão.

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Brandão, F., Plenio, M. Entanglement theory and the second law of thermodynamics. Nature Phys 4, 873–877 (2008). https://doi.org/10.1038/nphys1100

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