  The purity of quantum states is a key requirement for many quantum
applications. Improving the purity is limited by fundamental laws of
thermodynamics. Here we are probing the fundamental limits for a natural
approach to this problem, namely heat-bath algorithmic cooling(HBAC). The
existence of the cooling limit for HBAC techniques was proved by Schulman et
al. in, the limit however remained unknown for the past decade. Here for the
first time we find this limit. In the context of quantum thermodynamics, this
corresponds to the maximum extractable work from the quantum system.
