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85015797318
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note
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While it would be interesting to extend our proof to the hash-ing protocol, we note that for noisy local operations the hash-ing protocol, which requires Alice and Bob to apply a large number of controlled-NOT operations in every distillation step, usually performs much worse that the recurrence protocols. The reason for this lies in the fact that the noise (i.e., information loss) introduced with every controlled-NOT operation accumulates and rapidly shatters the potential information that could ideally be gained from the parity measurement.
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85015791801
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note
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We owe the term private entanglement to Charles Bennett.
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38
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85015800108
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note
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We are grateful to C.H. Bennett for pointing out this possibility.
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85015731055
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note
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This will prevent Eve from making use of any possibly prearranged order of the pairs, which Alice and Bob are meant to follow when performing the distillation process.
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43
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note
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While, strictly speaking, this equality holds only for N → ∞, the subsequent arguments also hold for the exact but more complicated form of Eq. (16) for finite N.
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