Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2410.22684

Imaginary part of timelike entanglement entropy

Wu-zhong Guo, Jin Xu

Published Oct 30, 2024• 0 citations
arXivHTMLPDF

Abstract

In this paper, we explore the imaginary part of the timelike entanglement entropy. In the context of field theory, it is more appropriate to obtain the timelike entanglement entropy through the Wick rotation of the twist operators. It is found that, in certain special cases, the imaginary part of the timelike entanglement entropy is related to the commutator of the twist operator and its first-order temporal derivative. To evaluate these commutators, we employ the operator product expansion of the twist operators, revealing that the commutator is generally universal across most scenarios. However, in more general cases, the imaginary part of the timelike entanglement entropy proves to be more complex. We compute the commutator of the twist operators along with its higher-order temporal derivatives. Utilizing these results, we derive a modified formula for the imaginary part of the timelike entanglement entropy. Furthermore, we extend this formula to the case of strip subregion in higher dimensions. Our analysis shows that for the strip geometry, the imaginary part of the timelike entanglement entropy is solely related to the commutators of the twist operator and its first-order temporal derivative. The findings presented in this paper provide valuable insights into the imaginary part of timelike entanglement entropy and its physical significance.

DOI
10.48550/arxiv.2410.22684

Related papers

Imaginary part of timelike entanglement entropy

Jin Xu, Wu-zhong Guo

A bstract In this paper, we explore the imaginary part of the timelike entanglement entropy. In the context of field theory, it is more appropriate to obtain the timelike entanglement entropy through the Wick rotation of the twist operators. It is found that, in certain special cases, the imaginary part of the timelike entanglement entropy is related to the commutator of the twist operator and its first-order temporal derivative. To evaluate these commutators, we employ the operator product expansion of the twist operators, revealing that the commutator is generally universal across most scenarios. However, in more general cases, the imaginary part of the timelike entanglement entropy proves to be more complex. We compute the commutator of the twist operators along with its higher-order temporal derivatives. Utilizing these results, we derive a modified formula for the imaginary part of the timelike entanglement entropy. Furthermore, we extend this formula to the case of strip subregion in higher dimensions. Our analysis shows that for the strip geometry, the imaginary part of the timelike entanglement entropy is solely related to the commutators of the twist operator and its first-order temporal derivative. The findings presented in this paper provide valuable insights into the imaginary part of timelike entanglement entropy and its physical significance.

Journal of High Energy Physics•Feb 14, 2025•15 citations

Relation between time- and spacelike entanglement entropy

Wu-zhong Guo, Song He, Yu-Xuan Zhang

In this study, we establish a connection between time- and spacelike entanglement entropy. We show that timelike entanglement entropy is closely related to spacelike entanglement entropy and its temporal derivative. For a broad class of states, it can be uniquely determined by a linear combination of spacelike entanglement entropy and its first-order temporal derivative. This relation holds, for instance, in states conformally equivalent to the vacuum in two-dimensional conformal field theories. For more general states, we demonstrate that the relation can be constructed perturbatively. Our results suggest that timelike entanglement entropy is constrained by causality. Moreover, this relation provides a unified framework for time- and spacelike entanglement entropy, within which the imaginary component of timelike entanglement entropy can be understood as arising from the noncommutativity between the twist operator and its first-order temporal derivative.

arXiv:2402.00268•Physical review. D/Physical review. D.•Oct 9, 2025•10 citations

Timelike entanglement entropy

Kazuki Doi, Jonathan Harper, Ali Mollabashi +2 more

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.

arXiv:2302.11695•Journal of High Energy Physics•May 8, 2023•104 citations

Entanglement of general subregions in time-dependent states

Tao Liu, Wu-zhong Guo, Song He

A bstract We develop a unified framework for computing Rényi and entanglement entropies of arbitrary spacetime intervals in time-dependent states of (1 + 1)-dimensional conformal field theories. By combining the spacetime density matrix formalism with the replica method, we show that entanglement entropy is well defined for both spacelike and timelike separations. Applying this framework to global quenches prepared by boundary states and to local quenches generated by operator insertions, we obtain analytic expressions for the entanglement entropy in general spacetime configurations. The results reveal qualitative differences between spacelike and timelike intervals: the timelike entanglement entropy is time-independent in the global quench model, depends solely on the temporal separation, and universally exhibits a constant imaginary contribution. These features are naturally explained by a generalized quasiparticle picture in which entanglement is produced precisely when one worldline of each quasiparticle pair intersects the interval. Furthermore, we demonstrate that the linear sum rule relating time- and spacelike entanglement persists in both global and local quenches, indicating a broader universality of spacetime entanglement in real-time quantum dynamics.

Journal of High Energy Physics•Jul 9, 2026•1 citation

Temporal Entanglement from Twist Correlators in 2d Conformal Field Theory and Holography

Alice Bernamonti, Federico Galli, Michal P. Heller +2 more

We formulate timelike entanglement entropy and its Rényi extension in two-dimensional conformal field theory through the analytic continuation of replica twist correlators to time-ordered, timelike-separated insertions. This field-theoretic construction grounds and generalizes recent developments, and applies to temporal subregions of arbitrary extent. Within three-dimensional holography, the semiclassical boundary correlator identifies boundary-anchored complex geodesics as the relevant bulk saddles and selects the one with the smallest real part of the length. This provides a direct boundary derivation of the proposed complex extremal surface prescription and extends to Rényi index $n>1$, for which we explicitly construct the corresponding complex cosmic brane geometry in the vacuum. We develop these ideas in several representative settings, including locally and globally excited states and quantum operator quenches, making manifest the precise agreement between boundary twist correlator and bulk complex geodesic calculations. For AdS-Vaidya, our approach predicts a different result from earlier piecewise geodesic constructions, while reproducing the field theory answer. Across these examples, the operator ordering uniquely determines the imaginary part of the complex-valued entropy, which is quantized in units of $cπ/6$ and sensitive to the effective causal structure but not to the underlying dynamics.

hep-thCross-listedarXiv:2607.14012•Jul 16, 2026•0 citations

Holographic timelike entanglement entropy in non-relativistic theories

Mir Afrasiar, Jaydeep Kumar Basak, Dimitrios Giataganas

A bstract Timelike entanglement entropy is a complex measure of information that is holographically realized by an appropriate combination of spacelike and timelike extremal surfaces. This measure is highly sensitive to Lorentz invariance breaking. In this work, we study the timelike entanglement entropy in non-relativistic theories, focusing on theories with hyperscaling violation and Lifshitz-like spatial anisotropy. The properties of the extremal surfaces, as well as the timelike entanglement entropy itself, depend heavily on the symmetry-breaking parameters of the theory. Consequently, we show that timelike entanglement can encode, to a large extent, the stability and naturalness of the theory. Furthermore, we find that timelike entanglement entropy identifies Fermi surfaces either through the logarithmic behavior of its real part or, alternatively, via its constant imaginary part, with this constant value depending on the theory’s Lifshitz exponent. This provides a novel interpretation for the imaginary component of this pseudoentropy. Additionally, we examine temporal entanglement entropy, an extension of timelike entanglement entropy to Euclidean space, and provide a comprehensive discussion of its properties in these theories.

arXiv:2411.18514•Journal of High Energy Physics•May 23, 2025•14 citations
In this paper, we explore the imaginary part of the timelike entanglement entropy. In the context of field theory, it is more appropriate to obtain the timelike entanglement entropy through the Wick rotation of the twist operators. It is found that, in certain special cases, the imaginary part of the timelike entanglement entropy is related to the commutator of the twist operator and its first-order temporal derivative. To evaluate these commutators, we employ the operator product expansion of the twist operators, revealing that the commutator is generally universal across most scenarios. However, in more general cases, the imaginary part of the timelike entanglement entropy proves to be more complex. We compute the commutator of the twist operators along with its higher-order temporal derivatives. Utilizing these results, we derive a modified formula for the imaginary part of the timelike entanglement entropy. Furthermore, we extend this formula to the case of strip subregion in higher dimensions. Our analysis shows that for the strip geometry, the imaginary part of the timelike entanglement entropy is solely related to the commutators of the twist operator and its first-order temporal derivative. The findings presented in this paper provide valuable insights into the imaginary part of timelike entanglement entropy and its physical significance.
A bstract In this paper, we explore the imaginary part of the timelike entanglement entropy. In the context of field theory, it is more appropriate to obtain the timelike entanglement entropy through the Wick rotation of the twist operators. It is found that, in certain special cases, the imaginary part of the timelike entanglement entropy is related to the commutator of the twist operator and its first-order temporal derivative. To evaluate these commutators, we employ the operator product expansion of the twist operators, revealing that the commutator is generally universal across most scenarios. However, in more general cases, the imaginary part of the timelike entanglement entropy proves to be more complex. We compute the commutator of the twist operators along with its higher-order temporal derivatives. Utilizing these results, we derive a modified formula for the imaginary part of the timelike entanglement entropy. Furthermore, we extend this formula to the case of strip subregion in higher dimensions. Our analysis shows that for the strip geometry, the imaginary part of the timelike entanglement entropy is solely related to the commutators of the twist operator and its first-order temporal derivative. The findings presented in this paper provide valuable insights into the imaginary part of timelike entanglement entropy and its physical significance.
In this study, we establish a connection between time- and spacelike entanglement entropy. We show that timelike entanglement entropy is closely related to spacelike entanglement entropy and its temporal derivative. For a broad class of states, it can be uniquely determined by a linear combination of spacelike entanglement entropy and its first-order temporal derivative. This relation holds, for instance, in states conformally equivalent to the vacuum in two-dimensional conformal field theories. For more general states, we demonstrate that the relation can be constructed perturbatively. Our results suggest that timelike entanglement entropy is constrained by causality. Moreover, this relation provides a unified framework for time- and spacelike entanglement entropy, within which the imaginary component of timelike entanglement entropy can be understood as arising from the noncommutativity between the twist operator and its first-order temporal derivative.
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.
A bstract We develop a unified framework for computing Rényi and entanglement entropies of arbitrary spacetime intervals in time-dependent states of (1 + 1)-dimensional conformal field theories. By combining the spacetime density matrix formalism with the replica method, we show that entanglement entropy is well defined for both spacelike and timelike separations. Applying this framework to global quenches prepared by boundary states and to local quenches generated by operator insertions, we obtain analytic expressions for the entanglement entropy in general spacetime configurations. The results reveal qualitative differences between spacelike and timelike intervals: the timelike entanglement entropy is time-independent in the global quench model, depends solely on the temporal separation, and universally exhibits a constant imaginary contribution. These features are naturally explained by a generalized quasiparticle picture in which entanglement is produced precisely when one worldline of each quasiparticle pair intersects the interval. Furthermore, we demonstrate that the linear sum rule relating time- and spacelike entanglement persists in both global and local quenches, indicating a broader universality of spacetime entanglement in real-time quantum dynamics.
We formulate timelike entanglement entropy and its Rényi extension in two-dimensional conformal field theory through the analytic continuation of replica twist correlators to time-ordered, timelike-separated insertions. This field-theoretic construction grounds and generalizes recent developments, and applies to temporal subregions of arbitrary extent. Within three-dimensional holography, the semiclassical boundary correlator identifies boundary-anchored complex geodesics as the relevant bulk saddles and selects the one with the smallest real part of the length. This provides a direct boundary derivation of the proposed complex extremal surface prescription and extends to Rényi index n>1n>1n>1, for which we explicitly construct the corresponding complex cosmic brane geometry in the vacuum. We develop these ideas in several representative settings, including locally and globally excited states and quantum operator quenches, making manifest the precise agreement between boundary twist correlator and bulk complex geodesic calculations. For AdS-Vaidya, our approach predicts a different result from earlier piecewise geodesic constructions, while reproducing the field theory answer. Across these examples, the operator ordering uniquely determines the imaginary part of the complex-valued entropy, which is quantized in units of cπ/6cπ/6cπ/6 and sensitive to the effective causal structure but not to the underlying dynamics.
A bstract Timelike entanglement entropy is a complex measure of information that is holographically realized by an appropriate combination of spacelike and timelike extremal surfaces. This measure is highly sensitive to Lorentz invariance breaking. In this work, we study the timelike entanglement entropy in non-relativistic theories, focusing on theories with hyperscaling violation and Lifshitz-like spatial anisotropy. The properties of the extremal surfaces, as well as the timelike entanglement entropy itself, depend heavily on the symmetry-breaking parameters of the theory. Consequently, we show that timelike entanglement can encode, to a large extent, the stability and naturalness of the theory. Furthermore, we find that timelike entanglement entropy identifies Fermi surfaces either through the logarithmic behavior of its real part or, alternatively, via its constant imaginary part, with this constant value depending on the theory’s Lifshitz exponent. This provides a novel interpretation for the imaginary component of this pseudoentropy. Additionally, we examine temporal entanglement entropy, an extension of timelike entanglement entropy to Euclidean space, and provide a comprehensive discussion of its properties in these theories.