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

Matrix Product Density Operators: when do they have a local parent Hamiltonian?