Information geometry encoded in bulk geometry
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Springer
Received: March 3, Revised: May 7, Accepted: May 26, Published: June 16,
2020 2020 2020 2020
Asato Tsuchiya and Kazushi Yamashiro Department of Physics, Shizuoka University, 836 Ohya, Suruga-ku, Shizuoka 422-8529, Japan Graduate School of Science and Technology, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu 432-8011, Japan
E-mail: [email protected], [email protected] Abstract: We study how information geometry is described by bulk geometry in the gauge/gravity correspondence. We consider a quantum information metric that measures the distance between the ground states of a CFT and a theory obtained by perturbing the CFT. We find a universal formula that represents the quantum information metric in terms of back reaction to the AdS bulk geometry. Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence ArXiv ePrint: 2002.11365
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)107
JHEP06(2020)107
Information geometry encoded in bulk geometry
Contents 1
2 Information metric in field theory
2
3 Information metric as on-shell action
4
4 Back reaction to the AdS geometry
5
5 Vector field
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6 Tensor field
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7 Conclusion and discussion
13
A Ricci tensor and scalar curvature
14
1
Introduction
Emergence of space-time (geometry) is considered to play an essential role in constructing quantum theory of gravity. Indeed, it is observed in various contexts including the AdS/CFT correspondence or the gauge/gravity correspondence [1–3], where the bulk direction on the gravity side emerges as the scale of renormalization group on the field theory side [2–5]. This observation motivates one to reconstruct full bulk geometry from field theory. The Ryu-Takayanagi formula [6] gives a hint to this problem. It relates entanglement entropy of a region in space on which a field theory is defined to the area of a minimal surface in the bulk whose boundary agrees with that of the region. Thus, it gives a relationship between quantum information theory and bulk geometry. In this paper, to further gain insights into this problem, we consider information metric in quantum information theory other than entanglement entropy, and investigate how they are encoded in bulk geometry. We represent information metric in terms of back reaction to the AdS bulk geometry, which is determined by dynamics of gravity. The geometrical quantity associated with the information metric is local in the bulk direction, while the minimal surface associated with entanglement entropy is not. Information metrics have been studied in the context of the AdS/CFT correspondence in [7–15]. The authors of [7] considered a CFT and a theory that is obtained by perturbing the CFT by an operator and calculate an information metric that measures the distance between the ground states of these two theories. They examined a gravity dual of a filed theory that is obtained by gluing the above two theories and found that the information metric is represented by the vol
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