Entanglement dynamics and relaxation in a few qubit system interacting with random collisions

The dynamics of a single qubit interacting by a sequence of pairwise collisions with an environment consisting of just two more qubits is analyzed. Each collision is modeled in terms of a random unitary operator with a uniform probability distribution described by the uniform Haar measure. We show that the purity of the system qubit as well as the bipartite and the tripartite entanglement reach time averaged equilibrium values characterized by large instantaneous fluctuations.These equilibrium values are independent of the order of collision among the qubits. The relaxation to equilibrium is analyzed also in terms of an ensemble average of random collision histories. Such average allows for a quantitative evaluation and interpretation of the decay constants. Furthermore a dependence of the transient dynamics on the initial degree of entanglement between the environment qubits is shown to exist. Finally the statistical properties of bipartite and tripartite entanglement are analyzed.