Purification Complexity of Gaussian States

We seek to test the recent conjecture of Agón, Headrick, and Swingle that holographic subregion complexity according to 'complexity = action' is dual to the purification complexity of the dual mixed state. We test this indirectly by considering the ground state of a free scalar field theory regularized to a lattice and computing the purification complexity of the reduced state on a subset of the lattice sites. Even though this system is not holographic, the hope is that the divergence structure will be universal enough that we may meaningfully compare to the holographic results.