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arXiv:2609.28621

Local Vacuum Entanglement through Most Entangled Modes

T. Rick Perche, Patricia Ribes-Metidieri

quant-phgr-qchep-thSubmitted Sep 23, 2026• Announced Sep 25, 2026• 0 citations
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Abstract

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.

Comments
5 pages, 4 figures + appendices

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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 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.
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Vacuum entanglement is a fundamental feature of quantum field theory exhibiting rich structure that is not completely understood. Here, we provide a complete characterization of the entanglement between two bounded spacelike-separated regions in a (1+1)-dimensional free massless real scalar field. Employing Gaussian state methods, we analytically compute the logarithmic negativity and construct closed-form solutions for the localized modes carrying it, called negativity cores. These results deepen our understanding of quantum fields and suggest extensions to higher dimensions and fermionic fields.
The many-body entanglement between two finite (size-ddd) disjoint vacuum regions of noninteracting lattice scalar field theory in one spatial dimension, i.e., a ${({d}_{A}\ifmmode\times\else\texttimes\fi{}{d}_{B})}_{\mathrm{mixed}}$ Gaussian continuous variable system, is locally transformed into a tensor-product core of ${({1}_{A}\ifmmode\times\else\texttimes\fi{}{1}_{B})}_{\mathrm{mixed}}$ entangled pairs. Accessible entanglement within these core pairs exhibits an exponential hierarchy and as such identifies the structure of dominant region modes from which vacuum entanglement could be extracted into a spatially separated pair of quantum detectors. Beyond the core, the remaining modes of the halo are determined to be ABABAB separable in isolation, as well as separable from the core. However, state preparation protocols that distribute entanglement in the form of ${({1}_{A}\ifmmode\times\else\texttimes\fi{}{1}_{B})}_{\mathrm{mixed}}$ core pairs are found to require additional entanglement in the halo that is obscured by classical correlations. This inaccessible (bound) halo entanglement is found to mirror the accessible entanglement, but with a step behavior as the continuum is approached. It remains possible that alternate initialization protocols that do not utilize the exponential hierarchy of core-pair entanglement may require less inaccessible entanglement. Entanglement consolidation is expected to persist in higher dimensions and may aid classical and quantum simulations of asymptotically free gauge field theories, such as quantum chromodynamics.