Entanglement measures for causally connected subregions and holography

TL;DR

This paper explores entanglement measures for causally connected subregions in quantum field theory and holography, introducing timelike entanglement entropy and related holographic duals, with explicit computations and consistency checks.

Contribution

It introduces a framework for defining and computing entanglement measures between causally connected regions, including timelike entanglement entropy and a holographic dual, extending entanglement concepts beyond spacelike separation.

Findings

Timelike entanglement entropy can be explicitly computed in quantum field theory.

The timelike entanglement wedge cross section is generally positive in AdS3/CFT2.

Reflected entropy for timelike intervals matches twice the entanglement wedge cross section at leading order.

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.