Gravitation from Entanglement

We have argued that the geometry of spacetime can be thought of as arising from the entanglement structure of the quantum state in an appropriate factorization. To match our empirical experience of the world, this emergent spacetime should respond to emergent energy-momentum through Einstein’s equation of general relativity.

The basic insight is Jacobson’s notion of “entanglement equilibrium” [17], extended to the case where spacetime itself is emergent rather than postulated [11]. Consider a subsystem in Hilbert space, in a situation where the overall quantum state is in the vacuum. It is then reasonable to imagine that the subsystem is in entangle­ment equilibrium: a small perturbation leaves the entropy of the region unchanged to first order. If we divide the entanglement into a small-scale ultraviolet term that determines the spacetime geometry, and a longer-scale infrared term characterizing matter fields propagating within that geometry, the change in one kind of entropy must be compensated for by a corresponding change in the other,

Here the left-hand side represents a change in geometry, and can be related directly to the spacetime curvature. The right-hand side represents a matter perturbation, which can be related to the modular Hamiltonian of an emergent effective field theory on the background. At the linearized level (the weak-field limit), it can be shown that this relation turns into the 00 component of Einstein’s equation in the synchronous gauge,

If the overall dynamics are approximately Lorentz invariant (which they must be for this program to work, although it’s unclear how to achieve this at this time), demanding that this equation hold in any frame implies the full linearized Einstein’s equation, SG= 8nGST^_v.

There are a number of assumptions at work here, but it seems plausible that the spacetime dynamics familiar from general relativity can arise in an emergent space­time purely from generic features of the entanglement structure of the quantum state. Following our quantum-first philosophy, this would be an example of finding gravity within quantum mechanics, rather than quantizing a classical model for gravitation.

8