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Maximally entangled states--a resource for quantum informationprocessing--can only be shared through noiseless quantum channels, whereas inpractice channels are noisy. Here we ask: Given a noisy quantum channel, whatis the maximum attainable purity (measured by singlet fraction) of sharedentanglement for single channel use and local trace preserving operations? Wefind an exact formula of the maximum singlet fraction attainable for a qubitchannel and give an explicit protocol to achieve the optimal value. Theprotocol distinguishes between unital and nonunital channels and requires nolocal post-processing. In particular, the optimal singlet fraction is achievedby transmitting part of an appropriate pure entangled state, which is maximallyentangled if and only if the channel is unital. A linear function of theoptimal singlet fraction is also shown to be an upper bound on the distillableentanglement of the mixed state dual to the channel.

Entanglement sharing through noisy qubit channels: One-shot optimal singlet fraction