Open-system quantum dynamics with correlated initial states, not completely positive maps, and non-Markovianity

Dynamical A and B maps have been employed extensively by Sudarshan and co-workers to investigate open-system evolution of quantum systems. A canonical structure of the A map is introduced here. It is shown that this canonical A map enables us to investigate whether the dynamics is completely positive (CP) or not completely positive (NCP) in an elegant way and, hence, it subsumes the basic results on open-system dynamics. Identifying memory effects in open-system evolution is gaining increasing importance recently and, here, a criterion of non-Markovianity, based on the relative entropy of the dynamical state is proposed. The relative entropy difference of the dynamical system serves as a complementary characterization-though not related directly-to the fidelity difference criterion proposed recently. Three typical examples of open-system evolution of a qubit, prepared initially in a correlated state with another qubit (environment), and evolving jointly under a specific unitary dynamics-which corresponds to a NCP dynamical map-are investigated by employing both the relative entropy difference and fidelity difference tests of non-Markovianity. The two-qubit initial states are chosen to be (i) a pure entangled state, (ii) the Werner state, which exemplifies both entangled and separable states of qubits, depending on a real parameter, and (iii) a separable mixed state. Both the relative entropy and fidelity criteria offer a nice display of how non-Markovianity manifests itself in all three examples.