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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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## 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).

## Author information

Authors

### Contributions

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.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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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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