  We consider a quantum key expansion (QKE) protocol based on
entanglement-assisted quantum error-correcting codes (EAQECCs). In these
protocols, a seed of a previously shared secret key is used in the
post-processing stage of a standard quantum key distribution protocol like the
Bennett-Brassard 1984 protocol, in order to produce a larger secret key. This
protocol was proposed by Luo and Devetak, but codes leading to good performance
have not been investigated. We look into a family of EAQECCs generated by
classical finite geometry (FG) low-density parity-check (LDPC) codes, for which
very efficient iterative decoders exist. A critical observation is that almost
all errors in the resulting secret key result from uncorrectable block errors
that can be detected by an additional syndrome check and an additional sampling
step. Bad blocks can then be discarded. We make some changes to the original
protocol to avoid the consumption of the preshared key when the protocol fails.
This allows us to greatly reduce the bit error rate of the key at the cost of a
minor reduction in the key production rate, but without increasing the
consumption rate of the preshared key. We present numerical simulations for the
family of FG LDPC codes, and show that this improved QKE protocol has a good
net key production rate even at relatively high error rates, for appropriate
choices of these codes.
