Entanglement Structures in Quantum Field Theories: Negativity Cores and Bound Entanglement in the Vacuum

The many-body entanglement between two finite (size-d) disjoint vacuum regions of noninteracting lattice scalar field theory in one spatial dimension, i.e., a (dA×dB)mixed Gaussian continuous variable system, is locally transformed into a tensor-product core of (1A×1B)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 AB separable in isolation, as well as separable from the core. However, state preparation protocols that distribute entanglement in the form of (1A×1B)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.