  Compressed sensing (CS) demonstrates that sparse signals can be recovered
from underdetermined linear measurements. We focus on the joint sparse recovery
problem where multiple signals share the same common sparse support sets, and
they are measured through the same sensing matrix. Leveraging a recent
information theoretic characterization of single signal CS, we formulate the
optimal minimum mean square error (MMSE) estimation problem, and derive a
belief propagation algorithm, its relaxed version, for the joint sparse
recovery problem and an approximate message passing algorithm. In addition,
using density evolution, we provide a sufficient condition for exact recovery.
