  We design a universal quantum homogenizer, which is a quantum machine that
takes as an input a system qubit initially in the state $\rho$ and a set of N
reservoir qubits initially prepared in the same state $\xi$. In the homogenizer
the system qubit sequentially interacts with the reservoir qubits via the
partial swap transformation. The homogenizer realizes, in the limit sense, the
transformation such that at the output each qubit is in an arbitratily small
neighbourhood of the state $\xi$ irrespective of the initial states of the
system and the reservoir qubits. This means that the system qubit undergoes an
evolution that has a fixed point, which is the reservoir state $\xi$. We also
study approximate homogenization when the reservoir is composed of a finite set
of identically prepared qubits. The homogenizer allows us to understand various
aspects of the dynamics of open systems interacting with environments in
non-equilibrium states. In particular, the reversibility vs or irreversibility
of the dynamics of the open system is directly linked to specific (classical)
information about the order in which the reservoir qubits interacted with the
system qubit. This aspect of the homogenizer leads to a model of a quantum safe
with a classical combination.We analyze in detail how entanglement between the
reservoir and the system is created during the process of quantum
homogenization. We show that the information about the initial state of the
system qubit is stored in the entanglement between the homogenized qubits.
