Symmetric extension in two-way quantum key distribution

We introduce symmetric extensions of bipartite quantum states as a tool for analyzing protocols that distill secret key from quantum correlations. Whether the correlations are coming from a prepare-and-measure quantum key distribution scheme or from an entanglement-based scheme, the protocol has to produce effective states without a symmetric extension in order to succeed. By formulating the symmetric extension problem as a semidefinite program, we solve the problem for Bell-diagonal states. Applying this result to the six-state and BB84 schemes, we show that for the entangled states that cannot be distilled by current key distillation procedures, the failure can be understood in terms of a failure to break a symmetric extension.