Abstract
A sharp version of the information paradox involves a seeming violation of the monogamy of entanglement during black hole evaporation. We construct an analogous paradox in empty anti-de Sitter space. In a local quantum field theory, Bell correlations between operators localized in mutually spacelike regions are monogamous. We show, through a controlled calculation, that this property can be violated by an order-1 factor in a theory of gravity. This example demonstrates that what appears to be a violation of the monogamy of entanglement may just be a subtle violation of locality in quantum gravity.
Highlights
A sharp version of the information paradox involves a seeming violation of the monogamy of entanglement during black hole evaporation
Between different spatial regions are monogamous. We show that this monogamy is violated dramatically in a theory of quantum gravity, even in empty anti-de Sitter space
This produces a “paradox” that is analogous to the “cloning” and “monogamy” paradoxes for evaporating black-holes. This construction provides strong evidence that these paradoxes can be resolved by recognizing that degrees of freedom cannot be localized in quantum gravity and information that appears to be present in one region of space can be extracted from another region
Summary
We start by reviewing how the monogamy of entanglement constrains Bell correlations. Consider two pairs of operators {A1, A2} and {B1, B2} with operator norms, Ai , Bi ≤ 1 and with [Ai, Bj] = 0. A state that yields 2 < |〈CAB〉| ≤ 2 2 displays correlations between the operators Ai and Bi beyond classical correlations — which implies that the degrees of freedom probed by these operators are entangled. If the operators Ai are “entangled” with Bi their correlations with Ci must be less than the allowed classical limit: |〈CAB〉| > 2 =⇒ |〈CAC 〉| < 2. In a local quantum field theory, the criterion that Ai, Bi, Ci commute can be replaced by the statement that Ai, Bi, Ci are localized on spacelike separated regions. We will show that in a theory of gravity, if we consider operators Ai, Bi, Ci localized on spacelike separated regions, (2) is violated by an O (1) amount
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