  In canonical quantum gravity, the presence of spatial boundaries naturally
leads to a boundary quantum states, representing quantum boundary conditions
for the bulk fields. As a consequence, quantum states of the bulk geometry
needs to be upgraded to wave-functions valued in the boundary Hilbert space:
the bulk become quantum operator acting on boundary states. We apply this to
loop quantum gravity and describe spin networks with 2d boundary as
wave-functions mapping bulk holonomies to spin states on the boundary. This
sets the bulk-boundary relation in a clear mathematical framework, which allows
to define the boundary density matrix induced by a bulk spin network states
after tracing out the bulk degrees of freedom. We ask the question of the bulk
reconstruction and prove a boundary-to-bulk universal reconstruction procedure,
to be understood as a purification of the mixed boundary state into a pure bulk
state. We further perform a first investigation in the algebraic structure of
induced boundary density matrices and show how correlations between bulk
excitations, i.e. quanta of 3d geometry, get reflected into the boundary
density matrix.
