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Entanglement of pure states of bipartite quantum systems has been shown to have a unique measure in terms of the von Neumann entropy of the reduced states of either of its subsystems. The measure is established under entanglement manipulation of an asymptotically large number of copies of the bipartite pure state. In this paper, two different asymptotic measures of entanglement for arbitrary sequences of bipartite pure states are established. These are shown to coincide only when the sequence is information stable, in terms of the quantum spectral information rates of its corresponding sequence of subsystem states. Additional bounds on the optimal rates of entanglement manipulation protocols in quantum information theory are also presented. These include bounds given by generalizations of the coherent information bounds, Rains' bound, and the relative entropy of entanglement.