Upper bound on the secret key rate distillable from effective quantum correlations with imperfect detectors

We provide a simple method to obtain an upper bound on the secret key rate that is particularly suited to analyze practical realizations of quantum key distribution protocols with imperfect devices. We consider the so-called trusted device scenario where Eve cannot modify the actual detection devices employed by Alice and Bob. The upper bound obtained is based on the available measurements results, but it includes the effect of the noise and losses present in the detectors of the legitimate users.

Figure 1

Upper bound on the secret key rate $K_S$ given by Eq. (21) for the four-state (dashed) and the six-state (dash-dotted) QKD schemes. The classical correlated data $p_{ij}$ is obtained by measuring the quantum state ${\rho }_{AB}\left(e\right)=(1-2e)\mid {\psi }^{+}⟩⟨{\psi }^{+}\mid +e\mathbb{1}∕2$, where $e$ is the error probability of a depolarizing channel. The two upper lines (dashed and dash-dotted) correspond to the ideal case where the total dark count probability is $d=0$, and the detector efficiency is $\eta =1$. The two lines below represent the typical case in current realizations of QKD: $d={10}^{-6}$ and $\eta =0.15$. The solid line of the graphic represent the upper bound for $K_S$ given by the regularized relative entropy of entanglement $E_r^{\infty }\mathbf{\left(}{\rho }_{AB}\right(e\left)\mathbf{\right)}$. Note that we assume an asymmetric basis choice to suppress the sifting effect [36].