  We consider the problem of reconstructing global quantum states from local
data. Because the reconstruction problem has many solutions in general, we
consider the reconstructed state of maximum global entropy consistent with the
local data. We show that unique ground states of local Hamiltonians are exactly
reconstructed as the maximal entropy state. More generally, we show that if the
state in question is a ground state of a local Hamiltonian with a degenerate
space of locally indistinguishable ground states, then the maximal entropy
state is close to the ground state projector. We also show that local
reconstruction is possible for thermal states of local Hamiltonians. Finally,
we discuss a procedure to certify that the reconstructed state is close to the
true global state. We call the entropy of our reconstructed maximum entropy
state the "reconstruction entropy", and we discuss its relation to emergent
geometry in the context of holographic duality.
