Evolution and decoherence in finite level systems

Abstract

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: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. Ó 2002 Published by Elsevier Science Ltd. Stochastic dynamics of open systemsGiven an isolated quantum system with a finitely many dimensional state space represented by an n  n density matrix q with properties x à r q rs x s P 0; trðqÞ ¼ 1; q y ¼ q the dynamical evolution is by a suitable unitary transformationIf there were a Hamiltonian H which is time independent U ðt 2 ; t 1 Þ ¼ expðÀiHtÞ:If the Hamiltonian were time dependent this is modified to readwhere T is the time ordering. If the system is interacting with external fields which are constant or time dependent, this result obtains. All that is needed is that the external fields do not manifest any back reaction. If we were to take into account the back reaction this is no longer possible. The time evolution is still linear. The most general form of such a dependence is by a dynamical map [1]:* Tel

Completely positive maps and classical correlations

“…If the initial state of the system and its environment is simply separable (product) so that ρ SE = η ⊗ τ , then the eigenvalues of the dynamical map are all positive for any choice of unitary evolution [8,9]. In this case we can define…”

Entanglement increase from local interactions and not completely positive maps

“…The interactions could be with large reservoirs instead of qubit controls [2,3,4]. Each qubit A or B could interact with a stream of reservoir qubits [15]. Here we are interested in the way the entanglement is changed by the combination of the two interactions.…”

General symmetry in the reduced dynamics of two-level system

“…The parametrization of completely positive maps was discussed in Ref. [13]. The structure and parametrization of not completely positive maps was later understood in Ref.…”