Evolution and decoherence in finite level systems

Forms of dynamics of open finite level systems is formulated. We give a presentation of stochastic dynamics of such systems in terms of maps. Completely positive maps are classified and parametrized. If the system is coupled to a companion system, the contraction of the unitary evolution of the combined system leads to a completely positive map of the system density matrix. The inverse problem of embedding a stochastic map in a unitary map of the combined system is posed and solved. This construction is not unique. Dephasing and decoherence stem from the same mechanism. The decoherence induced may be viewed in terms of the distance between the diagonal forms of the initial density matrix and the final: D=∑iλ′i−λi. The unitary factor of the evolution does not contribute to the decoherence so defined. A model is given for the stochastic evolution of a qubit coupled to a qubit. This yields a completely positive map for the original qubit. A triangle inequality constraint obtains for the three relaxation times; and it is due to the complete positivity of the map. Some comments are made about bipartite entangled systems in relation to maps.