  We investigate the possibility of a Markovian quantum master equation (QME)
that consistently describes a finite-dimensional system, a part of which is
weakly coupled to a thermal bath. In order to preserve complete positivity and
trace, such a QME must be of Lindblad form. For physical consistency, it should
additionally preserve local conservation laws and be able to show
thermalization. We search of Lindblad equations satisfying these additional
criteria. First, we show that the microscopically derived Bloch-Redfield
equation (RE) violates complete positivity unless in extremely special cases.
We then prove that imposing complete positivity and demanding preservation of
local conservation laws enforces the Lindblad operators and the lamb-shift
Hamiltonian to be `local', i.e, to be supported only on the part of the system
directly coupled to the bath. We then cast the problem of finding `local'
Lindblad QME which can show thermalization into a semidefinite program (SDP).
We call this the thermalization optimization problem (TOP). For given system
parameters and temperature, the solution of the TOP conclusively shows whether
the desired type of QME is possible up to a given precision. Whenever possible,
it also outputs a form for such a QME. For a XXZ chain of few qubits, fixing a
reasonably high precision, we find that such a QME is impossible over a
considerably wide parameter regime when only the first qubit is coupled to the
bath. Remarkably, we find that when the first two qubits are attached to the
bath, such a QME becomes possible over much of the same paramater regime,
including a wide range of temperatures.
