  We investigate which bipartite quantum states admit a symmetric extension and
apply the results in the analysis of noise thresholds in quantum key
distribution protocols with two-way postprocessing.
  We find that states that admit a symmetric extension can be decomposed into a
convex combination of states that admit a_pure_ symmetric extension. For a
state to have a pure symmetric extension, its nonzero eigenvalues must be equal
to the nonzero eigenvalues of one of the subsystems. While this does not fully
characterize the set of states with a pure symmetric extension in all
dimensions, it is a full characterization for two qubits. A conjectured formula
characterizing two-qubit states with a (mixed or pure) symmetric extension is
presented. Proofs are given for some special classes. For multi-qubit
Bell-diagonal states, the symmetric extension is simplified using the
symmetries of the states and formulated as a semidefinite program.
  Quantum key distribution with two-way advantage distillation steps is then
analyzed. For the commonly used repetition code advantage distillation, the
failure to distill a secret key beyond the currently known noise thresholds is
explained by a failure to break a symmetric extension. More general linear
advantage distillation schemes are analyzed, but for those schemes analyzed, we
find that they tolerate strictly less noise than repetition code advantage
distillation before they fail to break a symmetric extension. The results
suggest that no other linear advantage distillation scheme can reach the noise
threshold of 27.6 % for the six-state protocol which is achieved by repetition
code advantage distillation.
