Spacetime from Hilbert Space
Fortunately, we are guided in our quest by the fact that we know a great deal about what an appropriate emergent description should look like—a local effective field theory defined on a semiclassical four-dimensional dynamical spacetime.
The first step is to choose a decomposition of the Hilbert space HR (representing, for example, the interior of our cosmic horizon) into finite-dimensional micro-factors. We can say that the Hamiltonian is “local” with respect to such a decomposition if, for some small integer k, the Hamiltonian connects any specific factor Hat to no more than k other factors; intuitively, this corresponds to the idea that degrees of freedom at one location only interact with other degrees of freedom nearby.It turns out that a generic Hamiltonian will not be local with respect to any decomposition, and for the special Hamiltonians that can be written in a local form, the decomposition in which that works is essentially unique [9]. In other words, for the right kind of Hamiltonian, there is a natural decomposition of Hilbert space in which physics looks local, which is fixed by the spectrum alone. From the empirical success of local quantum field theory, we will henceforth assume that the Hamiltonian of the world is of this type, at least for low-lying states near the vacuum.
This preferred local decomposition naturally defines a graph structure on the space of Hilbert-space factors, where each node corresponds to a factor and two nodes are connected by an edge if they have a nonzero interaction in the Hamiltonian. To go from this topological structure to a geometric one, we need to look beyond the Hamiltonian to the specifics of an individual low-lying state. Given any factor of Hilbert space constructed from a collection of smaller factors, we can construct its density matrix and entropy,
and given any two such factors HA and we can define their mutual information 
Guided again by what we know about quantum field theory, we consider “redundancy- constrained” states, which capture the notion that nearby degrees of freedom are highly entangled, while faraway ones are unentangled.
The mutual information allows us to assign weights to the various edges in our Hilbert-space-factor graph. With an appropriate choice of weighting, these weights can be interpreted as distances, with large mutual information corresponding to short distances [10]. That gives our graph an emergent spatial geometry, from which we can find a best-fit smooth manifold using multidimensional scaling (Alternatively, the entropy across a surface can be associated with the surface’s area, and the emergent geometry defined using a Radon transform [11]). As the quantum state evolves with time according to the Schrodinger equation, the spatial geometry does as well; interpreting these surfaces as spacelike slices with zero extrinsic curvature yields an entire spacetime with a well-defined geometry.
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