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 ${(d_A\times d_B)}_{\mathrm{mixed}}$ Gaussian continuous variable system, is locally transformed into a tensor-product core of ${(1_A\times 1_B)}_{\mathrm{mixed}}$ 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 $AB$ separable in isolation, as well as separable from the core. However, state preparation protocols that distribute entanglement in the form of ${(1_A\times 1_B)}_{\mathrm{mixed}}$ 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.

Figure 1

Diagrammatic representation of consolidating two regions of the vacuum into a core of accessible entanglement and halo with inaccessible entanglement. Prior to consolidation (left), the two regions are entangled with multimode ${(d\times d)}_{\mathrm{mixed}}$ entanglement. After consolidation through local transformation, the negativity is captured in a core of ${(1\times 1)}_{\mathrm{mixed}}$ pairs that manifest the exponential hierarchy in entanglement contributions (Sec. 3b). The teal dashed lines indicate two partitions that are individually, but not simultaneously, separable.

Figure 2

Logarithmic negativity contributions for entanglement between vacuum regions of the one-dimensional massive scalar field with $d=30$ and $m=0.003$ as a function of partially transposed symplectic eigenvalue index for a series of separations $\tilde{r}$. The inset shows the same values with scaled $y$ components as $y^{\prime }=-\sqrt{-\ln y}$ such that linear behavior indicates Gaussian decay structure.

Figure 3

Logarithmic negativity quantifying core (accessible, dashed lines) and halo (inaccessible, solid lines) entanglement between two disjoint regions of the latticized one-dimensional massive ($m=0.003/d$) scalar field vacuum as a function of dimensionless separation $\tilde{r}/d$, where $\tilde{r}$ is the integer separation in lattice units and $d$ is the pixelated size of each region. The inset diagrammatically reproduces the solid lines of the main panel, vertically displaced to enhance visibility of emergent structure.