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Note that this does not exclude that if a violation of this inequality is observed, one can also obtain a violation of one of the inequalities in Eq. (3) large enough for full distillability. However, we have some numerical evidence suggesting that this is not the case.
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Note that once singlets between the sender and the other N - 1 parties are prepared, they can establish pairwise perfectly correlated keys and then run classical secret sharing protocols.
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