A supply of Werner pairs each has singlet fraction

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Problem

A supply of Werner pairs each has singlet fraction fs=0.80f_{\rm s}=0.80. One round of recurrence distillation acts on two such pairs at a time; by the recurrence map, the output singlet fraction and the probability of keeping the surviving pair are fs′=fs2+19(1−fs)2p,p=fs2+23fs(1−fs)+59(1−fs)2.f_{\rm s}'=\frac{f_{\rm s}^2+\tfrac19(1-f_{\rm s})^2}{p},\qquad p=f_{\rm s}^2+\tfrac23f_{\rm s}(1-f_{\rm s})+\tfrac59(1-f_{\rm s})^2. Calculate pp and fs′f_{\rm s}' for this input.

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