Time evolution of Two-qubit Entanglement

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We show that the entanglement dynamics for a closed two-qubit system is part of a 10-dimensional complex linear differential equation defined on a supersphere, and the coefficients therein are completely determined by the Hamiltonian. We apply the result to investigate two physical examples of Josephson junction qubits and exchange Hamiltonians, deriving analytic solutions for the time evolution of entanglement. The Hamiltonian coefficients determines whether the entanglement is periodic. These results allow of investigating how to generate and manipulate entanglements efficiently, which are required by both quantum computation and quantum communication.