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Physical review. D/Physical review. D.

Multimode nature of spacetime entanglement in QFT

Iván Agulló, Béatrice Bonga, Eduardo Martín-Martínez, Sergi Nadal-Gisbert, Tales Rick Perche, José Polo-Gómez, Patricia Ribes-Metidieri, Bruno de S. L. Torres

Published Apr 7, 2025• 6 citations
Publisher

Abstract

We demonstrate the presence of multimode entanglement in the vacuum state of a free, massless scalar quantum field in four-dimensional flat spacetime between two sets of field modes, each contained within a spacetime region that is causally disconnected from the other. This is true despite the fact that entanglement between pairs of individual field modes is sparse and appears only when the two individual modes are carefully selected. Our results reveal that, while entanglement between individual modes is limited, bipartite multimode entanglement in quantum field theory is ubiquitous. We further argue that such multimode entanglement is operationally extractable, and it forms the basis of the entanglement commonly discussed in entanglement harvesting protocols.

DOI
10.1103/physrevd.111.085013

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We demonstrate the presence of multimode entanglement in the vacuum state of a free, massless scalar quantum field in four-dimensional flat spacetime between two sets of field modes, each contained within a spacetime region that is causally disconnected from the other. This is true despite the fact that entanglement between pairs of individual field modes is sparse and appears only when the two individual modes are carefully selected. Our results reveal that, while entanglement between individual modes is limited, bipartite multimode entanglement in quantum field theory is ubiquitous. We further argue that such multimode entanglement is operationally extractable, and it forms the basis of the entanglement commonly discussed in entanglement harvesting protocols.
We demonstrate the presence of multimode entanglement in the vacuum state of a free, massless scalar quantum field in four-dimensional flat spacetime between two sets of field modes, each contained within a spacetime region that is causally disconnected from the other. This is true despite the fact that entanglement between pairs of individual field modes is sparse and appears only when the two individual modes are carefully selected. Our results reveal that, while entanglement between individual modes is limited, bipartite multimode entanglement in quantum field theory is ubiquitous. We further argue that such multimode entanglement is operationally extractable, and it forms the basis of the entanglement commonly discussed in entanglement harvesting protocols.
It is well known that entanglement is widespread in quantum field theory, in the following sense: every Reeh-Schlieder state contains entanglement between any two spatially separated regions. This applies, in particular, to the vacuum of a noninteracting scalar theory in Minkowski spacetime. Discussions on entanglement in field theory have focused mainly on subsystems containing infinitely many degrees of freedom---typically, the field modes that are supported within a compact region of space. In this article, we study entanglement in subsystems made of finitely many field degrees of freedom, in a free scalar theory in D+1D+1D+1-dimensional Minkowski spacetime. The focus on finitely many modes of the field is motivated by the finite capabilities of real experiments. We find that entanglement between finite-dimensional subsystems is not common at all, and that one needs to carefully select the support of modes for entanglement to show up. We also find that entanglement is increasingly sparser in higher dimensions. We conclude that entanglement in Minkowski spacetime is significantly less ubiquitous than normally thought.
We find the shape of the modes containing most of the entanglement between two disjoint spherical regions in the vacuum of a free massless real scalar field in 1+3 Minkowski spacetime. Using the Klco-Beck-Savage method we approximate the field and momentum profile of the vacuum most entangled modes (MEMs) between two spheres. We define a non-perturbative mode-swap entanglement harvesting protocol where two spacelike separated probes extract the maximal amount of entanglement between two regions. Our results are a first step in the direction of understanding the local entanglement structure of quantum fields, pinpointing the exact degrees of freedom that encode accessible quantum correlations in QFT.
We revisit the entanglement harvesting protocol when two detectors are in causal contact. We study the role of field-mediated communication in generating entanglement between the two detectors interacting with a quantum field. We provide a quantitative estimator of the relative contribution of communication versus genuine entanglement harvesting. For massless scalar fields in flat spacetime, we show that when two detectors can communicate via the field, the detectors do not really harvest entanglement from the field, and instead they get entangled only via the field-mediated communication channel. In other words, in these scenarios the entanglement harvesting protocol is truly ``harvesting entanglement'' from the field only when the detectors are not able to communicate. In contrast, for massive scalar fields both communication and genuine harvesting contribute equally to the bipartite entanglement when the detectors are causally connected. These results emphasize the importance of taking into account the causal relationships between two parties involved in this relativistic quantum information protocol before we can declare that it is truly entanglement harvesting.
First, we study the entanglement entropy S AB of a d + 1 dimensional massless free scalar field on two disjoint compact spatial regions A and B whose sizes are R 1 and R 2 , respectively, and the distance between them is r.The state of the massless free scalar field is the vacuum state.We consider the mutual information S A;B ≡ S A + S B -S AB where S A and S B are the entanglement entropy on the inside region of A and B, respectively.We develop the computational method based on that of Bombelli et al and obtain the result thatWhen A and B are spheres in d = 2 and 3, we obtain explicit resultsBy using the method used in the Minkowski spacetime case with some modifications, we study the entanglement entropy, S C , of the massless free scalar field on the outside region C of two black holes A and B whose radii are R 1 and R 2 and how it depends on the distance, r( R 1 , R 2 ) , between two black holes.If we can consider the entanglement entropy as thermodynamic entropy, we can see the entropic force (we call this entanglement entropic force) acting on the two black holes from the r dependence of S C .We consider the property of the entropic force.
Quantum fields in vacuum states carry an infinite amount of quantum entanglement, and its entanglement entropy has ultraviolet divergence. Harvesting protocols of the vacuum entanglement have been investigated, but its efficiency is very low to date. The main reason for the low efficiency originates from the fact that the extracted entanglement is embedded in a mixed state of two external devices. We propose a general protocol with high efficiency by extracting pure state entanglement from the field. We only use bi-linear interactions between the fields and external devices. Even though the ultraviolet cutoff remains finite, the protocol is capable of extracting a huge amount of entanglement. Hence an infinite amount of entanglement extraction is attained without the ultraviolet divergence of the field entanglement entropy in a continuum limit. There exists a trade-off relation between the extracted entanglement and its energy cost.