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Relation between timelike and spacelike entanglement entropy

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

TL;DR

This work establishes a precise link between timelike and spacelike entanglement entropies in 2D CFTs by showing that timelike entanglement entropy can be expressed as a linear combination of spacelike entanglement entropy and its first-order time derivative on the Cauchy surface $t=0$, with the imaginary part arising from a twist-derivative commutator. The authors derive the result explicitly for the vacuum state and demonstrate its validity for broader classes of states, including thermal states and AdS$_3$-dual holographic states, under a holographic RT framework. They analyze the conditions under which the relation holds, discuss its interpretation via pseudo entropy and twist operator dynamics, and outline extensions requiring additional operator contributions in more general OPE scenarios. The work suggests a pathway to generalize timelike EE beyond the vacuum and connects timelike EE to existing entropic frameworks in quantum field theory and holography.mathematical notation is consistently presented in $...$ throughout.

Abstract

In this study, we establish a connection between timelike and spacelike entanglement entropy. Specifically, for a diverse range of states, the timelike entanglement entropy is uniquely determined by a linear combination of the spacelike entanglement entropy and its first-order temporal derivative. This framework reveals that the imaginary component of the timelike entanglement entropy primarily originates from the non-commutativity between the twist operator and its first-order temporal derivative. Furthermore, we analyze the constraints of this relation and highlight the possible extension to accommodate more complex state configurations.

Relation between timelike and spacelike entanglement entropy

TL;DR

This work establishes a precise link between timelike and spacelike entanglement entropies in 2D CFTs by showing that timelike entanglement entropy can be expressed as a linear combination of spacelike entanglement entropy and its first-order time derivative on the Cauchy surface , with the imaginary part arising from a twist-derivative commutator. The authors derive the result explicitly for the vacuum state and demonstrate its validity for broader classes of states, including thermal states and AdS-dual holographic states, under a holographic RT framework. They analyze the conditions under which the relation holds, discuss its interpretation via pseudo entropy and twist operator dynamics, and outline extensions requiring additional operator contributions in more general OPE scenarios. The work suggests a pathway to generalize timelike EE beyond the vacuum and connects timelike EE to existing entropic frameworks in quantum field theory and holography.mathematical notation is consistently presented in throughout.

Abstract

In this study, we establish a connection between timelike and spacelike entanglement entropy. Specifically, for a diverse range of states, the timelike entanglement entropy is uniquely determined by a linear combination of the spacelike entanglement entropy and its first-order temporal derivative. This framework reveals that the imaginary component of the timelike entanglement entropy primarily originates from the non-commutativity between the twist operator and its first-order temporal derivative. Furthermore, we analyze the constraints of this relation and highlight the possible extension to accommodate more complex state configurations.
Paper Structure (14 sections, 83 equations, 1 figure)

This paper contains 14 sections, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Derivation of the relation for vacuum state
  3. Application to more general states
  4. The significance of the relation
  5. Extension of the relation
  6. Discussion
  7. Appendix
  8. Appendix A: Canonical quantization of two-dimensional massless free theory
  9. Method 1:
  10. Method 2:
  11. Appendix B: The relation (\ref{['linearcombination']}) for thermal state
  12. Appendix C: States dual to pure AdS$_3$
  13. Appendix D: Imaginary part of timelike entanglement entropy
  14. Appendix E: OPE of EE for general cases

Figures (1)

  • Figure 1: (a) An subregion $A$ on the Cauchy surface $t=0$ and its causal domain $\mathcal{D}_A$. (b) In a typical scenario where $(t,x)$ and $(t',x')$ are timelike, their past light cones intersect with four points at $t=0$ with $-u<-u'<v'<v$.