This talk will 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. For physical consistency, we will demand that the QME should preserve local conservation laws and be able to show thermalization. After providing some background on QMEs, I will present our three main results :

1. The microscopically derived Redfield equation (RE), which is known to preserve local conservation laws and show thermalization, necessarily violates complete positivity unless in extremely special cases. These special cases can be easily identified.
   
2. I will then turn to Lindblad QMEs and show 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.
   
3. Finally, I will show how the problem of finding `local' Lindblad QME which can show thermalization can be turned into a semidefinite program (SDP). This SDP can be solved numerically for any specific example, and its solution conclusively shows whether the desired type of QME is possible up to a given precision. Whenever a QME is possible, it also outputs a form for such a QME.
   

Taken together, our results indicate that the possibility of a Markovian QME with the desired properties must be taken on a case-by-case basis, since there are setups where such a QME is impossible. This is joint work with Manas Kulkarni, Abhishek Dhar and Archak Purkayastha. https://arxiv.org/abs/2301.02146.