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Entanglement of purification through holographic duality

Abstract

The gauge/gravity correspondence discovered two decades ago has had a profound influence on how the basic laws in physics should be formulated. In spite of the predictive power of holographic approaches (for example, when they are applied to strongly coupled condensed-matter physics problems), the fundamental reasons behind their success remain unclear. Recently, the role of quantum entanglement has come to the fore. Here we explore a quantity that connects gravity and quantum information in the light of the gauge/gravity correspondence. This is given by the minimal cross-section of the entanglement wedge that connects two disjoint subsystems in a gravity dual. In particular, we focus on various inequalities that are satisfied by this quantity. They suggest that it is a holographic counterpart of the quantity called entanglement of purification, which measures a bipartite correlation in a given mixed state. We give a heuristic argument that supports this identification based on a tensor network interpretation of holography. This predicts that the entanglement of purification satisfies the strong superadditivity for holographic conformal field theories.

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Fig. 1: Sketches of the entanglement wedge cross-section.
Fig. 2: The proof of a bound for the entanglement wedge cross-section.
Fig. 3: The proof of the strong superadditivity (18).
Fig. 4

References

  1. 1.

    Maldacena, J. M. The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys.2, 231–252 (1998).

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    Ryu, S. & Takayanagi, T. Holographic derivation of entanglement entropy from AdS/CFT. Phys. Rev. Lett.96, 181602 (2006).

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Hubeny, V. E., Rangamani, M. & Takayanagi, T. A covariant holographic entanglement entropy proposal. J. High. Energy Phys.0707, 062 (2007).

    ADS  MathSciNet  Article  Google Scholar 

  4. 4.

    Van Raamsdonk, M. Building up spacetime with quantum entanglement. Gen. Rel. Gravit.42, 2323–2329 (2010).

    ADS  MathSciNet  Article  Google Scholar 

  5. 5.

    Swingle, B. Entanglement renormalization and holography. Phys. Rev. D.86, 065007 (2012).

    ADS  Article  Google Scholar 

  6. 6.

    Miyaji, M. & Takayanagi, T. Surface/state correspondence as a generalized holography. Progr. Theor. Exp. Phys.2015, 073B03 (2015).

    Article  Google Scholar 

  7. 7.

    Pastawski, F., Yoshida, B., Harlow, D. & Preskill, J. Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence. J. High. Energy Phys.06, 149 (2015).

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    Hayden, P. et al. Holographic duality from random tensor networks. J. High. Energy Phys.11, 009 (2016).

    ADS  MathSciNet  Article  Google Scholar 

  9. 9.

    Freedman, M. & Headrick, M. Bit threads and holographic entanglement. Commun. Math. Phys.352, 407–438 (2017).

    ADS  MathSciNet  Article  Google Scholar 

  10. 10.

    Miyaji, M., Takayanagi, T. & Watanabe, K. From path integrals to tensor networks for AdS/CFT. Phys. Rev. D.95, 066004 (2017).

    ADS  MathSciNet  Article  Google Scholar 

  11. 11.

    Caputa, P., Kundu, N., Miyaji, M., Takayanagi, T. & Watanabe, K. Anti-de Sitter space from optimization of path integrals in conformal field theories. Phys. Rev. Lett.119, 071602 (2017).

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    Caputa, P., Kundu, N., Miyaji, M., Takayanagi, T. & Watanabe, K. Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT. J. High. Energy Phys.11, 097 (2017).

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    Headrick, M. & Takayanagi, T. A holographic proof of the strong subadditivity of entanglement entropy. Phys. Rev. D76, 106013 (2007).

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Hayden, P., Headrick, M. & Maloney, A. Holographic mutual information is monogamous. Phys. Rev. D87, 046003 (2013).

    ADS  Article  Google Scholar 

  15. 15.

    Headrick, M. General properties of holographic entanglement entropy. J. High. Energy Phys.03, 085 (2014).

    ADS  MathSciNet  Article  Google Scholar 

  16. 16.

    Bao, N. et al. The holographic entropy cone. J. High. Energy Phys.09, 130 (2015).

    ADS  MathSciNet  Article  Google Scholar 

  17. 17.

    Bengtsson, I. & Zyczkowski, K. Geometry of Quantum States: An Introduction to Quantum Entanglement (Cambridge Univ. Press, Cambridge, 2006).

  18. 18.

    Horodecki, R., Horodecki, P., Horodecki, M. & Horodecki, K. Quantum entanglement. Rev. Mod. Phys.81, 865–942 (2009).

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    Czech, B., Karczmarek, J. L., Nogueira, F. & Van Raamsdonk, M. The gravity dual of a density matrix. Class. Quant. Grav.29, 155009 (2012).

    ADS  MathSciNet  Article  Google Scholar 

  20. 20.

    Wall, A. C. Maximin surfaces, and the strong subadditivity of the covariant holographic entanglement entropy. Class. Quant. Grav.31, 225007 (2014).

    ADS  MathSciNet  Article  Google Scholar 

  21. 21.

    Headrick, M., Hubeny, V. E., Lawrence, A. & Rangamani, M. Causality and holographic entanglement entropy. J. High. Energy Phys.12, 162 (2014).

    ADS  Article  Google Scholar 

  22. 22.

    Headrick, M. Entanglement Rényi entropies in holographic theories. Phys. Rev. D.82, 126010 (2010).

    ADS  Article  Google Scholar 

  23. 23.

    Terhal, B. M., Horodecki, M., Leung, D. W. & DiVincenzo, D. P. The entanglement of purification. J. Math. Phys.43, 4286–4298 (2002).

    ADS  MathSciNet  Article  Google Scholar 

  24. 24.

    Bagchi, S. & Pati, A. K. Monogamy, polygamy, and other properties of entanglement of purification. Phys. Rev. A91, 042323 (2015).

    ADS  Article  Google Scholar 

  25. 25.

    Chen, J. & Winter, A. Non-additivity of the entanglement of purification (beyond reasonable doubt). Preprint at https://arxiv.org/abs/1206.1307 (2012).

  26. 26.

    DiVincenzo, D. P., Horodecki, M., Leung, D. W., Smolin, J. A. & Terhal, B. M. Locking classical correlation in quantum states. Phys. Rev. Lett.92, 067902 (2004).

    ADS  Article  Google Scholar 

  27. 27.

    Horodecki, K., Horodecki, M., Horodecki, P. & Oppenheim, J. Locking entanglement measures with a single qubit. Phys. Rev. Lett.94, 200501 (2005).

    ADS  Article  Google Scholar 

  28. 28.

    Christandl, M. & Winter, A. Uncertainty, monogamy, and locking of quantum correlations. IEEE Trans. Inf. Theory51, 3159–3165 (2005).

    MathSciNet  Article  Google Scholar 

  29. 29.

    Tucci, R. R. Entanglement of distillation and conditional mutual information. Preprint at https://arxiv.org/abs/quant-ph/0202144 (2002).

  30. 30.

    Christandl, M. & Winter, A. “Squashed entanglement”: An additive entanglement measure. J. Math. Phys.45, 829 (2004).

    ADS  MathSciNet  Article  Google Scholar 

  31. 31.

    Vidal, G. Entanglement renormalization. Phys. Rev. Lett.99, 220405 (2007).

    ADS  Article  Google Scholar 

  32. 32.

    Vidal, G. A class of quantum many-body states that can be efficiently simulated. Phys. Rev. Lett.101, 110501 (2008).

    ADS  Article  Google Scholar 

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Acknowledgements

We thank V. Hubeny, N. Kubo, N. Lashkari, T. Numasawa, H. Ooguri, J. Preskill, M. Rangamani, N. Shiba, T. Ugajin and G. Vidal for useful conversations. We are very grateful to J. Oppenheim for valuable comments on the properties of entanglement of purification. We also thank M. Headrick and H. Maxfield very much for helpful comments on the draft of this article, from which we learned that they had independent ideas on the entanglement wedge cross-section from different perspectives. T.T. is supported by the Simons Foundation through the `It from Qubit' collaboration and by JSPS Grant-in-Aid for Scientific Research (A) no.16H02182. T.T. is also supported by the World Premier International Research Center Initiative (WPI Initiative) from the Japan Ministry of Education, Culture, Sports, Science and Technology (MEXT).

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T.T. studied the holographic formula of entanglement wedge cross section and its tensor network interpretation. K.U. identified this with a holographic counterpart of entanglement of purification and showed that the holographic formula reproduces the correct inequalities.

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Correspondence to Tadashi Takayanagi.

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Umemoto, K., Takayanagi, T. Entanglement of purification through holographic duality. Nature Phys 14, 573–577 (2018). https://doi.org/10.1038/s41567-018-0075-2

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