Timelike entanglement entropy

Published for SISSA by Springer Received: March 8, 2023 Accepted: April 26, 2023 Published: May 8, 2023 Timelike entanglement entropy a Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan b Inamori Research Institute for Science, 620 Suiginya-cho, Shimogyo-ku, Kyoto 600-8411, Japan c Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8582, Japan E-mail: , , , , 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. Keywords: AdS-CFT Correspondence, Models of Quantum Gravity ArXiv ePrint: 2302.11695 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP05(2023)052 JHEP05(2023)052 Kazuki Doi,a Jonathan Harper,a Ali Mollabashi,a Tadashi Takayanagia,b,c and Yusuke Takia Contents 2 2 Timelike entanglement entropy in QFT 2.1 Definition via the replica method 2.2 Another equivalent definition via Wick rotation of coordinates 2.3 Timelike entanglement entropy of Dirac fermion on a torus 2.4 Numerical method 4 4 6 8 10 3 Holographic timelike entanglement entropy in AdS3 /CFT2 3.1 Pure AdS3 3.2 Comments on extreme surfaces in AdS3 3.3 BTZ black hole 3.3.1 Single interval 3.3.2 Timelike mutual information 3.3.3 Timelike thermofield mutual information 3.3.4 The necessity of timelike entanglement entropy in thermofield double state 3.4 BTZ with shock wave 3.5 Timelike EE under local quench in thermofield double state from the CFT 3.6 Rotating BTZ 16 16 18 20 22 24 27 4 Holographic timelike entanglement entropy in AdSd+1 /CFTd 4.1 Hyperbolic subsystem 4.1.1 Wick rotation 4.1.2 Extremal surface 4.1.3 Global coordinate 4.2 Strip subsystem 38 38 38 39 41 42 5 Holographic pseudo entropy in dS/CFT 5.1 Holographic pseudo entropy in dS3 /CFT2 5.2 Relation to timelike entanglement entropy 5.3 Higher dimensions 44 45 48 50 6 Conclusions and discussions 53 A Calculation of thermofield mutual information for holographic CFTs 55 –1– 29 30 34 36 JHEP05(2023)052 1 Introduction 1 Introduction The AdS/CFT correspondence [1] tells us that a space coordinate in an anti-de Sitter (AdS) spacetime can emerge from a conformal field theory (CFT). The mechanism of this emergent space can be described quantitatively by considering the holographic entanglement entropy (HEE) [2–4]. Specifically the entanglement entropy (EE) in a CFT is computed from the area of an extremal surface in AdS. The entanglement entropy SA for a subsystem A is defined by (1.1) ρA is the reduced density matrix obtained by tracing out the complement B of A: ρA = TrB ρtot , (1.2) where ρtot is the density matrix for the total system and the total Hilbert space is assumed to factorize as Htot = HA ⊗ HB . The holographic entanglement entropy computes SA from the area of an extremeal surface by SA = Area(ΓA ) . 4GN (1.3) This relation leads to the remarkable idea that the space coordinate in an AdS emerges from quantum entanglement [5, 6]. This raises a natural question: can the time coordinate also emerge from some quantum information theoretic property? To make progress in this problem, we first need to find a quantity which is directly related to the emergence of the time coordinate. Motivated by this, the main purpose of this paper is to introduce a new quantity called timelike entanglement entropy (timelike EE) and to study its properties. 1 (T) The timelike EE SA is defined by analytically continuing the standard EE to the case where the subsystem A is a timelike region. Indeed, in the AdS/CFT correspondence, the imaginary part of this quantity is related to the area of a timelike extremal surface, providing a generalization of (1.3), as depicted in left panel of figure 1. A part of these new results was briefly reported in the letter article [16], focusing on a few simple setups of AdS3 /CFT2 . Refer to [17–19] for independent works on timelike EE and see also [20–23]. In this full paper, we will present more general results of timelike EE in various setups of AdS/CFT. Moreover, in the context of dS/CFT [24], we can also see that the holographic entanglement entropy has contributions from timelike extremal surfaces in addition to a spacelike surface as depicted in the right panel of figure 1. Holography in de Sitter space (dS), so called the dS/CFT correspondence [24] argues that gravity on a de Sitter space is dual to a Euclidean CFT on its future infinity. In this holography, as opposed to AdS/CFT, the time coordinate emerges from the Euclidean CFT. Such a Euclidean CFT is expected to be exotic and non-unitary. A limited number of examples of CFTs dual to de Sitter spaces have been found in four dimensional higher spin gravity [25], in three dimensional Einstein 1 Refer to [7–15], for other ideas on how to study quantum entanglement in timelike setups. –2– JHEP05(2023)052 SA = −TrA [ρA log ρA ]. A t t A de Sitter Anti de Sitter gravity [26, 27] and in two dimensions [28, 29]. Since there is no spacelike geodesic between two distinct points on the dS boundary at future infinity, the holographic entropy becomes complex-valued [27, 30–34]. We point out that both the timelike entanglement entropy and the complex-valued holographic entropy in dS/CFT, can properly be interpreted as pseudo entropy, introduced in [35]. Pseudo entropy is defined as follows. Decomposing the total Hilbert space into those of subsystems A and B, we introduce the reduced transition matrix for two pure states |ψi and |ϕi, by |ψihϕ| . hϕ|ψi (1.4) SA = −Tr[τA log τA ]. (1.5)  τA = TrB  (P) Finally, pseudo entropy SA is defined by (P) Remarkably, in Euclidean time-dependent asymptotically AdS spaces, the pseudo entropy is (...truncated)