Abstract
We define a new complex-valued measure of information called the timelike entanglement entropy (EE) which in the boundary theory can be viewed as a Wick rotation that changes a spacelike boundary subregion to a timelike one. An explicit definition of the timelike EE in 2d field theories is provided followed by numerical computations which agree with the analytic continuation of the replica method for CFTs. We argue that timelike EE should be correctly interpreted as another measure previously considered, the pseudo entropy, which is the von Neumann entropy of a reduced transition matrix. Our results strongly imply that the imaginary part of the pseudo entropy describes an emergent time which generalizes the notion of an emergent space from quantum entanglement. For holographic systems we define the timelike EE as the total complex valued area of a particular stationary combination of both space and timelike extremal surfaces which are homologous to the boundary region. For the examples considered we find explicit matching of our optimization procedure and the careful implementation of the Wick rotation in the boundary CFT. We also make progress on higher dimensional generalizations and relations to holographic pseudo entropy in de Sitter space.
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Acknowledgments
We are grateful to Xi Dong, Daniel Harlow, Patrick Hayden, Juan Maldacena and Jonathan Oppenheim for useful comments on pseudo entropy and timelike entanglement entropy. This work is supported by the Simons Foundation through the “It from Qubit” collaboration and by MEXT KAKENHI Grant-in-Aid for Transformative Research Areas (A) through the “Extreme Universe” collaboration: Grant Number 21H05187. This work is also supported by Inamori Research Institute for Science and by JSPS Grant-in-Aid for Scientific Research (A) No. 21H04469. Y. T. is supported by Grant-in-Aid for JSPS Fellows No. 22J21950. TT would like to thank the 2022 Simons Collaboration on It from Qubit Annual Meeting where a part of the present work was presented. Some of the numerical calculations were carried out on Yukawa-21 at YITP in Kyoto University.
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Doi, K., Harper, J., Mollabashi, A. et al. Timelike entanglement entropy. J. High Energ. Phys. 2023, 52 (2023). https://doi.org/10.1007/JHEP05(2023)052
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DOI: https://doi.org/10.1007/JHEP05(2023)052