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Symmetric extension and quantum key distribution


I will talk about how to characterize states with a symmetric extension and why symmetric extensions are relevant to quantum key distribution, both with one-way and two-way post-processing.

A bipartite state shared between Alice and Bob is said to have a symmetric extension if it can be extended to a triparite state, such that the third party has a part of the state equivalent to Bob's.  For the case when Alice and Bob each holds a qubit, the characterization simplifies tremendously, and I present a conjectured simple formula which we have proven in special cases.  In higher dimension the characterization is necessarily more complicated, but we can still get necessary conditions.

For quantum key distribution with one-way postprocessing the implication of a symmetric extension is immediate: such a state is not useful.  By using two-way procedures in the postprocessing we can break symmetric extensions.  Still, a given two-way procedure can be tested to see for which states symmetric extension is actually broken. For example the failure of Gottesman and Lo's two-way procedure to distill key when the the QBER is above 20% and 27,6% for the BB84 protocol and 6-state protocol, can be explained by a failure to break a symmetric extension.