Randomly distilling W-class states into general configurations of two-party entanglement

In this article we obtain new results for the task of converting a \textit{single} $N$-qubit W-class state (of the form $\sqrt{x_0}\ket{00...0}+\sqrt{x_1}\ket{10...0}+...+\sqrt{x_N}\ket{00...1}$) into maximum entanglement shared between two random parties. Previous studies in random distillation have not considered how the particular choice of target pairs affects the transformation, and here we develop a strategy for distilling into \textit{general} configurations of target pairs. We completely solve the problem of determining the optimal distillation probability for all three qubit configurations and most four qubit configurations when $x_0=0$. Our proof involves deriving new entanglement monotones defined on the set of four qubit W-class states. As an additional application of our results, we present new upper bounds for converting a generic W-class state into the standard W state $\ket{W_N}=\sqrt{\frac{1}{N}}(\ket{10...0}+...+\ket{00...1})$.