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We explore the important fundamental question of how quantum information islocalized in quantum gravity, in a perturbative approach. Familiar descriptionsof localization of information, such as via tensor factorization of the Hilbertspace or a net of commuting subalgebras of operators, conflict with basicgravitational properties -- specifically gauge invariance -- already at leadingorder in perturbation theory. However, previous work found that information canbe classically localized in a region in a way such that measurements, includingthose of the gravitational field, outside the region are insensitive to thatinformation, and only measure total Poincare charges. This paper shows that,working to leading order in the gravitational coupling, a similar quantumresult holds, leading to a definition of a "gravitational splitting" on theHilbert space for gravity. Such localization of information also argues againsta role for "soft hair" in resolving the information problem for black holes.This basic mathematical structure plausibly plays a foundational role in thequantum description of gravity.

Gravitational splitting at first order: Quantum information localization in gravity