Energy cost of localization of relational quantum information

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