  Entanglement of spatially separated quantum states is usually defined with
respect to a reference frame provided by some external observer. Thus, if one
wishes to localize the quantum information within a spatially separated
entangled state, one must enact an entanglement extraction protocol also
defined with respect to that external frame. Entanglement extraction for
Gaussian ground states in such an external frame construction has been shown to
require a minimum energy and is hence an interesting process for gravitational
physics, where examinations of localization vs. energy cost have a long
history. General covariance however, precludes dependence on external frames.
In order to enact an extraction protocol in a generally covariant theory,
dependence on the external reference frame must first be removed and the states
made relational. We examine the implementation of an extraction protocol for
Gaussian states, who's center-of-mass and relational degrees of freedom are
entangled, in a relational toy model where translation invariance stands in for
full diffeomorphism invariance. Constructing fully relational states and the
corresponding extraction/localization can, in principle, be done in two ways.
External frame position information can be removed through $G$-twirling over
translations or one can spontaneously break the translation symmetry via the
gradient of an auxiliary field, or $Z$-model. We determine the energetics of
quantum information localization after the states have been made fully
relational via both the $G$-twirl and $Z$-model. We also show one can smoothly
transition between the two approaches via positive operator valued measurements
(POVM).
