Fundamental limitations in Lindblad descriptions of systems weakly coupled to baths

It is very common in the literature to write a Markovian quantum master equation in Lindblad form to describe a system with multiple degrees of freedom and weakly connected to multiple thermal baths which can, in general, be at different temperatures and chemical potentials. However, the microscopically derived quantum master equation up to leading order in a system-bath coupling is of the so-called Redfield form, which is known to not preserve complete positivity in most cases. Additional approximations to the Redfield equation are required to obtain a Lindblad form. We lay down some fundamental requirements for any further approximations to the Redfield equation, which, if violated, leads to physical inconsistencies such as inaccuracies in the leading order populations and coherences in the energy eigenbasis, violation of thermalization, and violation of local conservation laws at the nonequilibrium steady state. We argue that one or more of these conditions will generically be violated in all the weak system-bath-coupling Lindblad descriptions existing in the literature to our knowledge. As an example, we study the recently derived universal Lindblad equation and use these conditions to show the violation of local conservation laws due to inaccurate coherences but accurate populations in the energy eigenbasis. Finally, we exemplify our analytical results numerically in an interacting open quantum spin system.