  We consider a spatial analogue of the quantum error correction threshold.
Given individual time-independent subsystems in which quantum information is
coherent over sufficiently long lengths, we show how the information can be
kept coherent for arbitrarily long lengths by forming time-independent
composite systems. The subsystem coherence length exhibits threshold behavior.
When it exceeds a length ${\xi}_{th}$, meaningful information can be extracted
from the ground state of the composite system. Otherwise, the information is
garbled. The threshold transition implies that the parent Hamiltonian of the
ground state has gone from gapped to gapless. Ramifications of the construction
for PEPS and for adiabatic quantum computation are noted.
