Quantum separability of the vacuum for scalar fields with a boundary

Using the Green’s function approach we investigate separability of the vacuum state of a massless scalar field with a single Dirichlet boundary. Separability is demonstrated using the positive partial transpose criterion for effective two-mode Gaussian states of collective operators. In contrast to the vacuum energy, entanglement of the vacuum is not modified by the presence of the boundary.

Figure 1

We average fields over two tiny boxes $B$ and $B^{\prime }$ centered at $\vec{x}$ and ${\vec{x}}^{\prime }$, respectively. The Dirichlet boundary is at $z=0$.