• Open Access

Michal P. Heller1,2,*, Fabio Ori1,†, and Alexandre Serantes1,‡

• 1Department of Physics and Astronomy, Ghent University, 9000 Ghent, Belgium

• 2Institute of Theoretical Physics and Mark Kac Center for Complex Systems Research, Jagiellonian University, 30-348 Cracow, Poland

• *Contact author: michal.p.heller@ugent.be

• †Contact author: fabio.ori@ugent.be

• ‡Contact author: alexandre.serantesrubianes@ugent.be

Abstract

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.

• Gauge-gravity dualities

Popular Summary

Entanglement is one of the most fascinating features of quantum physics—it describes how parts of a system can remain deeply connected even when far apart. Scientists have long understood how to describe this connection in space, by dividing a system into spatial regions and measuring how much information they share. But what about connections that unfold in time, rather than space? In this study, we develop a consistent method for describing and calculating temporal entanglement, especially in the context of holographic theories that link gravity and quantum physics through the AdS/CFT correspondence.

To obtain a consistent definition of entanglement entropy in time, we start from the usual definition of spatial entanglement and gradually transform the region of interest so that it stretches across time. More specifically, in the holographic picture—where the entropy of quantum systems is represented by surfaces in a higher-dimensional space—we track the surfaces that define entanglement as they evolve from the spacelike case into the timelike one. When several possible surfaces appear, our method univocally identifies the one with the smallest real area as the correct physical choice.

Our approach eliminates the puzzling ambiguity about how to describe temporal entanglement when multiple surfaces appear. It also ensures that the new definition remains consistent with what we already know from the better-understood case of entanglement in space. This new perspective opens the door to studying how information is shared not just across space, but across time—offering new insights into how the notion of entanglement can be extended to the time direction.

Article Text

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