  The electromagnetic trapping of ion chains can be regarded as a process of
non-trivial entangled quantum state preparation within Hilbert spaces of the
local axial motional modes. To begin uncovering properties of this entanglement
resource produced as a byproduct of conventional ion-trap quantum information
processing, the quantum continuous-variable formalism is herein utilized to
focus on the leading-order entangled ground state of local motional modes in
the presence of a quadratic trapping potential. The decay of entanglement
between disjoint subsets of local modes is found to exhibit features of
entanglement structure and responses to partial measurement reminiscent of the
free massless scalar field vacuum. With significant fidelities between the two,
even for large system sizes, a framework is established for initializing
quantum field simulations via "imaging" extended entangled states from natural
sources, rather than building correlations through deep circuits of few-body
entangling operators. By calculating probabilities in discrete Fock subspaces
of the local motional modes, considerations are presented for locally
transferring these pre-distributed entanglement resources to the qudits of ion
internal energy levels, improving this procedure's anticipated experimental
viability.
