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Entanglement theory and the second law of thermodynamics

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.

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  • 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.

References

  1. Callen, H. B. Thermodynamics and an Introduction to Thermostatistics (Wiley, 1985).

    MATH  Google Scholar 

  2. Giles, R. Mathematical Foundations of Thermodynamics (Pergamon, 1964).

    MATH  Google Scholar 

  3. Lieb, E. H. & Yngvason, J. The physics and mathematics of the second law of thermodynamics. Phys. Rep. 310, 1–96 (1999).

    Article  ADS  MathSciNet  Google Scholar 

  4. Lieb, E. H. & Yngvason, J. Current Developments in Mathematics, 2001 Vol. 89 (International Press, 2002).

    Google Scholar 

  5. Plenio, M. B. & Virmani, S. An introduction to entanglement measures. Quantum. Inf. Comput. 7, 1–51 (2007).

    MathSciNet  MATH  Google Scholar 

  6. Horodecki, R., Horodecki, P., Horodecki, M. & Horodecki, K. Quantum entanglement. Rev. Mod. Phys. (in the press); preprint at <http://arxiv.org/abs/quant-ph/0702225> (2007).

  7. Bennett, C. H. & Divincenzo, D. P. Quantum information and computation. Nature 404, 247–255 (2000).

    Article  ADS  Google Scholar 

  8. Einstein, A., Podolsky, B. & Rosen, N. Can quantum mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935).

    Article  ADS  Google Scholar 

  9. Schrödinger, E. Die gegenwärtige Situation in der Quantenmechanik. Naturwissenschaften 23, 807–812 (1935).

    Article  ADS  Google Scholar 

  10. Bennett, C. H. et al. Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70, 1895–1899 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  11. Bennett, C. H., Bernstein, H. J., Popescu, S. & Schumacher, B. Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046–2052 (1996).

    Article  ADS  Google Scholar 

  12. Horodecki, M., Horodecki, P. & Horodecki, R. Mixed-state entanglement and distillation: Is there a bound entanglement in nature? Phys. Rev. Lett. 80, 5239–5242 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  13. Vidal, G. & Cirac, J. I. Irreversibility in asymptotic manipulations of entanglement. Phys. Rev. Lett. 86, 5803–5806 (2001).

    Article  ADS  Google Scholar 

  14. Yang, D., Horodecki, M., Horodecki, R. & Synak-Radtke, B. Irreversibility for all bound entangled states. Phys. Rev. Lett. 95, 190501 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  15. Popescu, S. & Rohrlich, D. Thermodynamics and the measure of entanglement. Phys. Rev. A 56, R3319–R3321 (1997).

    Article  ADS  MathSciNet  Google Scholar 

  16. Horodecki, P., Horodecki, R. & Horodecki, M. Entanglement and thermodynamical analogies. Acta Phys. Slov. 48, 141–156 (1998).

    MATH  Google Scholar 

  17. Plenio, M. B. & Vedral, V. Entanglement in quantum information theory. Contemp. Phys. 39, 431–466 (1998).

    Article  ADS  Google Scholar 

  18. Vedral, V. & Plenio, M. B. Entanglement measures and purification procedures. Phys. Rev. A 57, 1619–1633 (1998).

    Article  ADS  Google Scholar 

  19. Horodecki, M., Oppenheim, J. & Horodecki, R. Are the laws of entanglement theory thermodynamical? Phys. Rev. Lett. 89, 240403 (2002).

    Article  ADS  Google Scholar 

  20. Werner, R. F. Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277–4281 (1989).

    Article  ADS  Google Scholar 

  21. Vidal, G. & Tarrach, R. Robustness of entanglement. Phys. Rev. A 59, 141–155 (1999).

    Article  ADS  MathSciNet  Google Scholar 

  22. Harrow, A. W. & Nielsen, M. A. Robustness of quantum gate in the presence of noise. Phys. Rev. A 68, 012308 (2003).

    Article  ADS  Google Scholar 

  23. Vedral, V., Plenio, M. B., Rippin, M. A. & Knight, P. L. Quantifying entanglement. Phys. Rev. Lett. 78, 2275–2279 (1997).

    Article  ADS  MathSciNet  Google Scholar 

  24. Hiai, F. & Petz, D. The proper formula for relative entropy and its asymptotics in quantum probability. Commun. Math. Phys. 143, 99–114 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  25. Ogawa, T. & Nagaoka, H. Strong converse and Stein’s lemma in the quantum hypothesis testing. IEEE Trans. Inf. Theory 46, 2428–2433 (2000).

    Article  MathSciNet  Google Scholar 

  26. Renner, R. Symmetry of large physical systems implies independence of subsystems. Nature Phys. 3, 645–649 (2007).

    Article  ADS  Google Scholar 

  27. Eisert, J. & Plenio, M. B. Introduction to the basics of entanglement theory in continuous-variable systems. Int. J. Quant. Inf. 1, 479–506 (2003).

    Article  Google Scholar 

  28. Gour, G. & Spekkens, R. W. The resource theory of quantum reference frames: manipulation and monotones. New. J. Phys. 10, 033023 (2008).

    Article  ADS  Google Scholar 

  29. Maurer, U. Secret key agreement by public discussion from common information. IEEE Trans. Inf. Theory 39, 733–742 (1993).

    Article  MathSciNet  Google Scholar 

  30. Jonathan, D. & Plenio, M. B. Entanglement-assisted local manipulation of pure quantum states. Phys. Rev. Lett. 83, 3566–3569 (1999).

    Article  ADS  MathSciNet  Google Scholar 

  31. Brandão, F. G. S. L., Horodecki, M., Plenio, M. B. & Virmani, S. Remarks on the equivalence of full additivity and monotonicity for the entanglement cost. Open Syst. Inf. Dyn. 14, 333–339 (2007).

    Article  MathSciNet  Google Scholar 

  32. Brandão, F. G. S. L. & Plenio, M. B. A reversible theory of entanglement and its relation to the second law. Preprint at <http://arxiv.org/abs/0710.5827> (2007).

  33. Donald, M. J. & Horodecki, M. Continuity of relative entropy of entanglement. Phys. Lett. A 264, 257–260 (1999).

    Article  ADS  MathSciNet  Google Scholar 

  34. Rains, E. M. A semidefinite program for distillable entanglement. IEEE Trans. Inf. Theory 47, 2921–2933 (2001).

    Article  MathSciNet  Google Scholar 

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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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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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