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arXiv:2508.05158 · Physical review. D/Physical review. D.

Entanglement measures for causally connected subregions and holography

Xiangkun Gong, Wu-zhong Guo, Jin Xu

Published Apr 20, 2026• 2 citations
arXivHTMLPDF

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 A B 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 A B 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 admit well-defined extensions of other entanglement measures beyond the spacelike setting. With this motivation we study the analytic continuation of the twist correlators for reflected-entropy and propose a holographic geometric analog in complexified bulk geometry, providing evidence that entanglement measures can be extended to timelike regions.

DOI
10.1103/771p-4rkf

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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 A B 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 A B 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 admit well-defined extensions of other entanglement measures beyond the spacelike setting. With this motivation we study the analytic continuation of the twist correlators for reflected-entropy and propose a holographic geometric analog in complexified bulk geometry, providing evidence that entanglement measures can be extended to timelike regions.
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.
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.
Recently, several notions of entanglement in time have emerged as a novel frontier in quantum many-body physics, quantum field theory, and gravity. We propose a systematic prescription to characterize temporal entanglement in relativistic quantum field theory in a general state for an arbitrary subregion on a flat, constant time slice in a flat spacetime. Our prescription starts with the standard entanglement entropy of a spatial subregion and amounts to transporting the unchanged subregion to boosted time slices all the way across the light cone when it becomes, in general, a complex characterization of the corresponding temporal subregion. For holographic quantum field theories, our prescription amounts to an analytic continuation of all codimension-two bulk extremal surfaces satisfying the homology constraint and picking the one with the smallest real value of the area as the leading saddle point. We implement this prescription for holographic conformal field theories in thermal states on both a two-dimensional Lorentzian cylinder and three-dimensional Minkowski space, and we show that it leads to results with self-consistent physical properties of temporal entanglement.
We study various entanglement measures associated with certain nonconformal field theories. We consider nonconformal Dp\mathrm{D}pDp-brane backgrounds, which are dual to these field theories, for our holographic analysis. Restricting our interests in p=1p=1p=1, 2, 4, we explicitly compute properties of holographic entanglement entropy and the entanglement wedge cross section, EW{\mathrm{E}}_{W}EW​, corresponding to two parallel strip shaped boundary subregions in these setups. We study low and high temperature behaviors of these quantities analytically as well as by using numerical methods. In all cases, the EW{\mathrm{E}}_{W}EW​ decreases monotonically with temperature. We observe discontinuous jumps in EW{\mathrm{E}}_{W}EW​, while the width of (as well as the separation between) the subregions reaches critical values in all the cases considered. However, the corresponding holographic mutual information IM{\mathtt{I}}_{M}IM​ continuously decreases to zero for the aforementioned configurations. We also notice that the conjectured inequality ${\mathrm{E}}_{W}\ensuremath{\ge}{\mathtt{I}}_{M}/2$ still holds for nonconformal field theories as well. We analytically determine the critical separation between these subregions that triggers a phase transition in the holographic mutual information.
We develop a unified framework for computing Rényi and entanglement entropies of arbitrary spacetime intervals in time-dependent states of (1+1)(1+1)(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 propose a novel family of entanglement measures for time-separated subsystems. Our definitions are applicable to any quantum system, continuous or discrete. To illustrate their utility, we derive upper and lower bounds on time-separated correlation functions, akin to the bound on spatially separated correlators in terms of the mutual information. In certain cases our bounds are tight. For relativistic quantum field theories our definition agrees with the analytic continuation from spacelike to timelike separated regions. We provide relevant measurement protocols and execute them on the IBM quantum device ibm_sherbrooke for a simple qubit system. Also we perform explicit computations for an Ising spin chain, free fermions, (1+1)-dimensional conformal field theories and holographic theories. Finally we explain how the proposed entanglement in time provides a microscopic definition for the recently introduced timelike pseudoentropy.