  Pure quantum states play a central role in applications of quantum
information, both as initial states for many algorithms and as resources for
quantum error correction. Preparation of highly pure states that satisfy the
threshold for quantum error correction remains a challenge, not only for
ensemble implementations like NMR or ESR but also for other technologies.
Heat-Bath Algorithmic Cooling is a method to increase the purity of set of
qubits coupled to a bath. We investigated the achievable polarization by
analyzing the state when no more entropy can be extracted. In particular we
give an analytic form for the maximum polarization of the purified qubit and
corresponding state of the whole system for the case when the initial state of
the qubits is totally mixed. It is however possible to reach higher
polarization while starting with other states with higher polarization, thus
our result provides an achievable lower bound. We also give an upper bound of
the number of steps needed to get a certain required polarization.
