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 arbitrarily small neighborhood 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 nonequilibrium states. In particular, the reversibility vs 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.