Exact MEMs for two spherical regions in 1+3 dimensions

Determine the exact most entangled modes (MEMs), including their field-amplitude and momentum smearing profiles, for two disjoint spherical regions of a massless scalar field in 1+3-dimensional Minkowski spacetime.

Background

The paper numerically approximates the most entangled modes between two disjoint spherical regions by applying the entanglement-partner construction to finite-dimensional subspaces spanned by Laplacian eigenfunctions. The resulting field-amplitude smearing develops increasingly fine boundary oscillations and does not converge in the ordinary L² norm, whereas the momentum smearing exhibits convergent behavior. The authors compare these results with closed-form MEMs known for two intervals in 1+1 dimensions, whose field-amplitude profiles are not square-integrable but nevertheless produce finite smeared correlations and logarithmic negativity.

A closed-form characterization of the corresponding exact MEMs in 1+3 dimensions is not provided. Establishing these modes would clarify the continuum limit of the numerical construction, the boundary singularity structure of the field-amplitude profile, and the precise functional framework in which the associated smeared observables and entanglement measures are defined.

References

While the exact MEMs for two spherical regions in 1+3 dimensions are not known, closed-form expressions for the exact MEMs between two intervals of a massless scalar field in 1+1 dimensions have been found in.

— Local Vacuum Entanglement through Most Entangled Modes  (2609.28621 - Perche et al., 23 Sep 2026) in Appendix, Section “Regularity and Convergence of the MEMs and their Entanglement,” subsection “Insights from the 1+1-dimensional case”