Quantum key distribution protocols can be based on
quantum error correcting codes, where the structure of the code determines the
post processing protocol applied to a raw key produced by BB84 or a similar
scheme. Luo and Devetak showed that
basing a similar protocol on entanglement-assisted quantum error-correcting
codes (EAQECCs) leads to quantum key expansion (QKE) protocols, where some
amount of previously shared secret key is used as a seed in the post-processing
stage to produce a larger secret key. One of the promising aspects of EAQECCs
is that they can be constructed from classical linear codes that don't satisfy
the dual-containing property, which among other things allows the use of low
density parity-check (LDPC) codes with girth greater than 4, for which the
iterative decoding algorithm has better performance. We looked into QKE based on a family of
EAQECCs generated by classical finite geometry (FG) LDPC codes. Very efficient iterative decoders exist for
these codes, and they were shown by Hsieh, Yen and Hau to produce quantum LDPC
codes that require very little entanglement.
We modify the original QKE protocol to detect bad code blocks without
the consumption of secret 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 net key
production rate, but without increasing the consumption rate of pre-shared
key. Numerical simulations for the
family of FG LDPC codes show that this improved QKE protocol has a good net key
production rate even at relatively high error rates, for appropriate code
choices.