What is the minimum energy required to move a unit of heat from the cold side of a system to the hot side?
One obvious lower bound is that it must be at least as much as the amount of work that can be done by a maximum efficiency heat engine when that heat moves back. So for example, for a system with the cold end at 0°C and the hot end at 50°C (323K), maximum heat engine efficiency is 50/323 = ~15%. By this logic, since 1J of heat moving from hot to cold side can perform at most 0.15J of work, therefore it must take at least 0.15J of energy to move 1J of heat back from the cold end to the hot end at these given temperatures. Assuming I made no mistakes here.
Is this the actual theoretical lower bound on the energy required, or is there a higher theoretical lower bound?