No-Go Theorem Behind The Limit Of The Heat-Bath Algorithmic Cooling

Algorithmic cooling techniques provide tools to increase the purity of quantum states. It is known that the cooling with these techniques is limited. However, the physical root of the limit is still unclear. Here we show that the unitarity of the compression operation imposes the cooling limit of heat-bath algorithmic cooling. Specifically, we prove that the unitarity of the compression operation does not allow increasing the purity beyond the maximum of the individual purities. We formalize the limitations imposed by the unitarity of the compression operation in two theorems. We then introduce an optimal cooling technique and show that without the limitations of the unitary operations the cooling technique would exceed the limit of heat-bath algorithmic cooling. We also show that the limitations of unitarity lead to the exact cooling limit of algorithmic cooling. This work opens up new avenues to understanding the limits of dynamic cooling.