  We present a theoretical framework to investigate quantum thermodynamic
processes under non-Markovian system-bath interactions on the basis of the
hierarchical equations of motion (HEOM) approach, which is convenient to carry
out numerically "exact" calculations. This formalism is valuable because it can
be used to treat not only strong system-bath coupling but also system-bath
correlation or entanglement, which will be essential to characterize the heat
transport between the system and quantum heat baths. Using this formalism, we
demonstrated an importance of the thermodynamic effect from the tri-partite
correlations (TPC) for a two-level heat transfer model and a three-level
autonomous heat engine model under the conditions that the conventional quantum
master equation approaches are failed. Our numerical calculations show that TPC
contributions, which distinguish the heat current from the energy current, have
to be take into account to satisfy the thermodynamic laws.
