Quantum entanglement transformations via local operations and classical communication

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The primary goals of this thesis are two-fold: i) calculating the optimal entanglement transformation rate between two multipartite pure states via stochastic local operations and classical communication (SLOCC), and ii) showing the properties of a common resource for a set of multi-partite pure state via local operations and classical communication (LOCC) or SLOCC. We introduce a notion of entanglement transformation rate to characterize the asymptotic comparability of two multi-partite pure entangled states under SLOCC. For two well known SLOCC inequivalent three-qubit states: Greenberger-Horne-Zeilinger (GHZ) state and W state, we show that the entanglement transformation rate from GHZ state to W state is exactly 1. 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. Then, we discuss the common resource for a set of pure states. We have completely solved the bipartite pure states case by explicitly constructing a unique optimal common resource state for any given set of states via LOCC. In the multi-partite setting, the general problem becomes quite complicated, and we focus on finding non-trivial common resources for the whole multi-partite state space of given dimensions. We show that |GHZ₃〉 = (1/√3|) (|000〉+|111〉+|222〉) is a nontrivial common resource for three-qubit systems via LOCC. We also obtain a number of interesting properties of non-trivial common resource states for two N-qubit pure states and multi-partite systems via SLOCC.
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