Abstract
We consider the emergence from quantum entanglement of spacetime geometry in a bulk region. For certain classes of quantum states in an appropriately factorized Hilbert space, a spatial geometry can be defined by associating areas along codimension-one surfaces with the entanglement entropy between either side. We show how Radon transforms can be used to convert this data into a spatial metric. Under a particular set of assumptions, the time evolution of such a state traces out a four-dimensional spacetime geometry, and we argue using a modified version of Jacobson's "entanglement equilibrium" that the geometry should obey Einstein's equation in the weak-field limit. We also discuss how entanglement equilibrium is related to a generalization of the Ryu-Takayanagi formula in more general settings, and how quantum error correction can help specify the emergence map between the full quantum-gravity Hilbert space and the semiclassical limit of quantum fields propagating on a classical spacetime.
Highlights
There has been considerable recent interest in the idea of deriving an emergent spacetime geometry from the entanglement properties of a quantum state [1,2,3,4,5,6]
There we investigated how a spatial metric could be derived from a quantum state using the mutual information between different factors in Hilbert space
Our interest here is dynamical rather than static: to model the universe as a quantum state evolving in Hilbert space, show how the geometry of spacetime can emerge from the entanglement features of such a state in an appropriate factorization, and derive Einstein’s equation in the semiclassical limit, an approach we label bulk entanglement gravity (BEG)
Summary
There has been considerable recent interest in the idea of deriving an emergent spacetime geometry from the entanglement properties of a quantum state [1,2,3,4,5,6]. Our interest here is dynamical rather than static: to model the universe as a quantum state evolving in Hilbert space, show how the geometry of spacetime can emerge from the entanglement features of such a state in an appropriate factorization, and derive Einstein’s equation in the semiclassical limit, an approach we label bulk entanglement gravity (BEG). Following Harlow [28], we argue that this can be done purely from the entanglement structure of the state or from the properties of a quantum error-correction code (QECC) It seems that quantum error correction properties can naturally provide a separation between geometric and matter d.o.f. Our framework, partly inspired by [29], uses the radon transform to tie together previous work on the thermodynamics of spacetime [16,19], AdS=CFT approaches to emergent gravity [4], and kinematic space [30].
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