  The connection between quantum state recovery and quantum conditional mutual
information (QCMI) is studied for the class of purely generated finitely
correlated states (pgFCS) of one-dimensional quantum spin chains. For a
tripartition of the chain into two subsystems separated by a buffer region, it
is shown that a pgFCS is an approximate quantum Markov chain, and stronger, may
be approximated by a quantum Markov chain in trace distance, with an error
exponentially small in the buffer size. This implies that, (1) a locally
corrupted state can be approximately recovered by action of a quantum channel
on the buffer system, and (2) QCMI is exponentially small in the size of the
buffer region. Bounds on the exponential decay rate of QCMI and examples of
quantum recovery channels are presented.
