  We study whether one can write a Matrix Product Density Operator (MPDO) as
the Gibbs state of a quasi-local parent Hamiltonian. We conjecture this is the
case for generic MPDO and give supporting evidences. To investigate the
locality of the parent Hamiltonian, we take the approach of checking whether
the quantum conditional mutual information decays exponentially. The MPDO we
consider are constructed from a chain of 1-input/2-output (`Y-shaped')
completely-positive maps, i.e., the MPDO have a local purification. We derive
an upper bound on the conditional mutual information for bistochastic channels
and strictly positive channels and show that it decays exponentially if the
correctable algebra of the channel is trivial. We also introduce a conjecture
on a quantum data processing inequality that implies the exponential decay of
the conditional mutual information for every Y-shaped channel with trivial
correctable algebra. We additionally investigate a close but nonequivalent
cousin: MPDO measured in a local basis. We provide sufficient conditions for
the exponential decay of the conditional mutual information of the measured
states and numerically confirm they are generically true for certain random
MPDO.
