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We investigate which bipartite quantum states admit a symmetric extension andapply the results in the analysis of noise thresholds in quantum keydistribution protocols with two-way postprocessing.  We find that states that admit a symmetric extension can be decomposed into aconvex combination of states that admit a_pure_ symmetric extension. For astate to have a pure symmetric extension, its nonzero eigenvalues must be equalto the nonzero eigenvalues of one of the subsystems. While this does not fullycharacterize the set of states with a pure symmetric extension in alldimensions, it is a full characterization for two qubits. A conjectured formulacharacterizing two-qubit states with a (mixed or pure) symmetric extension ispresented. Proofs are given for some special classes. For multi-qubitBell-diagonal states, the symmetric extension is simplified using thesymmetries of the states and formulated as a semidefinite program.  Quantum key distribution with two-way advantage distillation steps is thenanalyzed. For the commonly used repetition code advantage distillation, thefailure to distill a secret key beyond the currently known noise thresholds isexplained by a failure to break a symmetric extension. More general linearadvantage distillation schemes are analyzed, but for those schemes analyzed, wefind that they tolerate strictly less noise than repetition code advantagedistillation before they fail to break a symmetric extension. The resultssuggest that no other linear advantage distillation scheme can reach the noisethreshold of 27.6 % for the six-state protocol which is achieved by repetitioncode advantage distillation.

Symmetric extension of bipartite quantum states and its use in quantum key distribution with two-way postprocessing