Exact calculation of entanglement negativity for a 1+1D massless scalar field using phase space methods

Quantum fields exhibit a rich entanglement structure which is still not fully understood.

In this work, we study the entanglement structure of the vacuum state of a massless scalar field in (1+1)-dimensions -- a paradigmatic case for both high energy and condensed matter physics.

We fully characterize the entanglement negativity between two arbitrary compact spacelike-separated regions of the field by calculating the logarithmic negativity along with the modes carrying it, called negativity cores.

We achieve this using a framework based on the Kähler structure of Gaussian states, wherein we calculate the diagonalization of the operator associated with the partially-transposed restricted linear complex structure.

In doing so, we extend the methods of this framework by proposing a basis-independent definition of the transpose operation.

The explicit diagonalization we perform is enabled by a reformulation of the eigenvalue problem as a boundary value problem in the complex plane.

Our results also suggest extensions to higher dimensions and fermionic fields.