Key rates for quantum key distribution protocols with asymmetric noise

We consider the asymptotic key rates achieved in the simplest quantum key distribution protocols, namely, the BB84 and the six-state protocols when nonuniform noise is present in the system. We first observe that higher qubit error rates do not necessarily imply lower key rates. Second, we consider protocols with advantage distillation and show that it can be advantageous to use the basis with higher quantum bit error rate for the key generation. We then discuss the relation between advantage distillation and entanglement distillation protocols. We show that applying advantage distillation to a string of bits formed by the outcomes of measurements in the basis with a higher quantum bit error rate is closely connected to the two-to-one entanglement distillation protocol of Deutsch-Ekert-Jozsa-Macchiavello-Popescu-Sanpera [Phys. Rev. Lett. 77, 2818 (1996)]. Finally, we discuss the implications of these results for implementations of quantum key distribution.

Figure 1

Asymptotic key rate of the six-state protocol with minimum leakage information reconciliation as a function of the QBER in the $Y$ basis $Q_Y$ for the family of Bell-diagonal states with $Q_X=Q_Z=0.1$.

Figure 2

(a) Singlet fidelity and (b) entanglement of formation for the family of Bell-diagonal states specified by QBERs $(0.1,Q_Y,0.1)$. Both quantities decrease monotonically as $Q_Y$ increases in the range of possible values of $Q_Y,0\le Q_Y\le 0.2$.

Figure 3

Key rates for the six-state protocol with advantage distillation, given by Protocol I, for the family of states $Q_X=Q_Z=0.1$. Blue (dark gray) curve shows the key rate when the $Z$ basis is used for key generation, and the yellow (light gray) curve when the $Y$ basis is used for key generation.

Figure 4

Comparison of the key rates for the BB84 protocol with advantage distillation given by Protocol I when the $X$ and the $Z$ bases are used for the key generation. The blue (dark gray) dots represent the parameters for which higher key rates are achieved when $Z$ is the key generation basis. The yellow (light gray) dots represent the case when higher rates are achieved if $X$ is used for key generation. One can observe that, for almost all ranges of parameters, higher rates are obtained when the basis with higher QBER is used for key generation. This behavior only inverts for a small range of parameters, near the limiting region where positive key can no longer be extracted.

Figure 5

Comparison of the key rates for the BB84 protocol with an advantage distillation protocol that uses blocks of size 7. The blue (dark gray) dots represent the parameters for which higher key rates are achieved when the $Z$ basis is used for key generation. The yellow (light gray) dots represent the case when higher key rates are achieved with measurements in the $X$ basis. For this case, it is advantageous to always use the basis with higher QBER for key generation.

Figure 6

Key rates for the six-state protocol with the advantage distillation protocol introduced in Ref. [18] for the family of states $Q_X=Q_Z=0.1$. Blue (dark gray) curve illustrates the key rates when the $Z$ basis is used for the key generation rounds, and the yellow (light gray) curve when the $Y$ basis is used for the key generation rounds.