Entanglement measures for causally connected subregions and holography

TL;DR

This work formalizes entanglement in time by introducing the transition operator T_{AB} for causally connected subregions, enabling timelike entanglement entropy (EE) computations via the Schwinger–Keldysh real-time replica method. It develops a holographic program by analytically continuing Euclidean RT surfaces to construct timelike RT surfaces and introduces the timelike entanglement wedge and its cross section (EWCS), establishing a holographic link to timelike reflected entropy at leading order. The authors present concrete 1+1D examples and extend the analysis to AdS$_{d+1}$ backgrounds, including BTZ and higher-dimensional black holes, with numerical verification of the timelike RT surfaces. They also generalize entanglement measures such as negativity and reflected entropy to timelike configurations, discuss purification subtleties for non-Hermitian T_{AB}, and propose a broader framework for timelike holography and real-time entanglement structure.

Abstract

In this paper, we investigate entanglement for causally connected subregions $A$ and $B$ in quantum field theory and holography. Recent developments have established that a transition operator $T_{AB}$ can be well-defined for such subregions, which is generally non-Hermitian. By employing the Schwinger-Keldysh formalism and the real-time replica method, we show how to construct $T_{AB}$ and compute associated entanglement measures. In certain configurations, this leads to a notion of timelike entanglement entropy, for which we provide explicit quantum field theory computations and propose a holographic dual via analytic continuation from the Euclidean setup. Both analytical and numerical results are compared and found consistent. If entanglement between causally connected subregions is to be meaningful, it should also be able to define other entanglement measures. Motivated by the spacelike case, we propose a timelike extension of the entanglement wedge cross section, though we do not expect it to carry the same physical interpretation. In AdS$_3$/CFT$_2$, we compute explicit examples and find that the timelike entanglement wedge cross section is generally positive. Furthermore, we show that the reflected entropy for timelike intervals -- obtained via analytic continuation of twist correlators -- coincides with twice the timelike entanglement wedge cross section at leading order in $G$, supporting a holographic duality in the timelike case. We also discuss the extension of other entanglement measures, such as logarithmic negativity, to timelike separated regions using replica methods. We highlight conceptual challenges in defining reflected entropy via canonical purification for non-Hermitian operators.

Figures (24)

• Figure 1: An illustration of subsystems $A$ and $B$: (left) $A$ and $B$ are causally disconnected; (right) $A$ and $B$ are causally connected.
• Figure 2: An illustration of the four possible configurations of regions $A$ and $B$: (a) $A$ and $B$ are causally disconnected; (b) $A$ and $B$ are Cauchy surfaces at different times; (c) $A$ lies entirely within the causal domain of $B$; (d) a portion of $B$ lies within the causal domain of $A$.
• Figure 3: A typical Keldysh contour.
• Figure 4: The Schwinger-Keldysh formalism illustrating the path-integral representation of ${\mathop\textrm{Tr}} \rho$. (a) Calculate ${\mathop\textrm{Tr}} \rho(t)$ by (\ref{['Tr-rho-singleU']}). (b) Calculate ${\mathop\textrm{Tr}} \rho(t_1)$ by (\ref{['Tr-rho-doubleU']}).
• Figure 5: The Schwinger-Keldysh formalism illustrating the path-integral representation of the reduced density matrix $\rho_A(t)$.