Obtaining a $W$ State from a Greenberger-Horne-Zeilinger State via Stochastic Local Operations and Classical Communication with a Rate Approaching Unity

We introduce a notion of the entanglement transformation rate to characterize the asymptotic comparability of two multipartite pure entangled states under stochastic local operations and classical communication (SLOCC). For two well known SLOCC inequivalent three-qubit states $|\mathrm{GHZ}⟩=(1/\sqrt{2})(|000⟩+|111⟩)$ and $|W⟩=(1/\sqrt{3})(|100⟩+|010⟩+|001⟩)$, we show that the entanglement transformation rate from $|\mathrm{GHZ}⟩$ to $|W⟩$ is exactly 1. That means that we can obtain one copy of the $W$ state from one copy of the Greenberg-Horne-Zeilinger (GHZ) state by SLOCC, asymptotically. We then apply similar techniques to obtain a lower bound on the entanglement transformation rates from an $N$-partite GHZ state to a class of Dicke states, and prove the tightness of this bound for some special cases which naturally generalize the $|W⟩$ state. A new lower bound on the tensor rank of the matrix permanent is also obtained by evaluating the tensor rank of Dicke states.