We present a detailed study of entanglement and decoherence in systems undergoing Markov and non-Markov interactions. The Markov approximation (shorthand for Born-Markov) makes it possible to find solutions for the dynamics of complicated open systems. We review this subject for the spin-boson and collisional decoherence models, as well as an exactly solvable special case. The Markov approximation requires very weak and infrequent interactions, which is not always feasible.;Merkli, Sigal, and Berman [[Merkli, 2008]] have developed a framework called resonance theory which can be used to calculate the reduced density matrix elements of an N-level system interacting with a massless bosonic environment at non-zero temperature to arbitrary precision in the interaction strength lambda. For a qubit system, they calculated the dynamics to O(lambda2) in the exponents, which happen to be the same as given by the Markov approximation. They calculated only one O(lambda2) contribution to the coefficients, which is non-Markov, using a method independent of resonance theory which involves the expansion of the joint thermal state of the system and environment at infinite time. We calculate all leading O(lambda 2) coefficients, which are non-Markov, and correct the O(lambda 2) term calculated by Merkli et al., which we obtain by two different methods, (Sections 4.3.7-4.3.9).;We also examine a system of two qubits undergoing independent decoherence under Markov interactions and give a new classification of their entanglement sudden death (ESD) behavior for arbitrary initial conditions (Section 5.3.1). Taking into account the non-Markov contributions obtained in Sections 4.3.7-4.3.9, we find that ESD can be delayed or quickened (Section 5.3.2). Such information is useful for the control of ESD since entanglement is a valuable resource for quantum computation.
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