  A notion of localization of information within quantum subsystems plays a key
role in describing the physics of quantum systems, and in particular is a
prerequisite for discussing important concepts such as entanglement and
information transfer. While subsystems can be readily defined for finite
quantum systems and in local quantum field theory, a corresponding definition
for gravitational systems is significantly complicated by the apparent
nonlocality arising due to gauge invariance, enforced by the constraints. A
related question is whether "soft hair" encodes otherwise localized
information, and the question of such localization also remains an important
puzzle for proposals that gravity emerges from another structure such as a
boundary field theory as in AdS/CFT. This paper describes different approaches
to defining local subsystem structure, and shows that at least classically,
perturbative gravity has localized subsystems based on a split structure,
generalizing the split property of quantum field theory. This, and related
arguments for QED, give simple explanations that in these theories there is
localized information that is independent of fields outside a region, in
particular so that there is no role for "soft hair" in encoding such
information. Additional subtleties appear in quantum gravity. We argue that
localized information exists in perturbative quantum gravity in the presence of
global symmetries, but that nonperturbative dynamics is likely tied to a
modification of such structure.
