Preprocessing noise in finite-size quantum key distribution

Abstract

It is known that preprocessing noise may boost quantum key distribution by expanding the range of values of tolerated noise. For BB84, adding trusted noise may allow the generation of secret keys even for qubit error rate (QBER) beyond the 11% threshold in the asymptotic regime. Here we study the effect of preprocessing noise in the finite-size regime where only a limited number of signals are exchanged between Alice and Bob. We compute tight numerical lower bounds in terms of the sandwiched Rényi entropy of order alpha, optimized via a two-step Frank-Wolfe algorithm, in the presence of a trusted flipping probability q. We find that trusted noise improves the key rate only for a finite interval of alpha, from the alpha -> 1 limit up to alpha approx 1.4. By optimizing on the value of alpha, we determine finite-size key rates for different values of the QBER, observing enhancement due to trusted noise both in asymptotic and finite-size regimes. Finally, we determine the maximum tolerable QBER as a function of the block size.

Figures (5)

• Figure 1: Schematic of BB84 protocol with trusted noise preprocessing, in the entanglement-based representation. Alice and Bob's classical bits obtained though qubit measurements are indicated as $b_A$ and $b_B$ respectively. The symbol $b'_B$ indicates Bob's bit after trusted bit flip.
• Figure 2: Largest absolute difference between secret-key rates with optimal trusted noise level $r(\alpha,p,q^*)$ and without trusted randomization procedure $r(\alpha, p, q=0)$, as in Eq. (\ref{['Delta_r']}), plotted vs the Rényi order $\alpha$. The level $p$ of external noise at which $\Delta r$ is found is displayed above the data points.
• Figure 3: Optimal trusted noise level $q^*(\alpha,p)$, as a function of the intrinsic eavesdropper's noise $p$ and Rényi order $\alpha$. The domain ($\alpha$,$p$) is restricted to those points for which $r(\alpha,p,q^*)>0$; the forbidden domain is colored in red. The optimal trusted flipping probability grows as $p>0.10$ for every value of $\alpha$ considered.
• Figure 4: Key rates at finite block size, following the optimization over the Rényi order $\alpha$. We comparison the key rate without trusted noise, and the key rate achieved with the optimal value of the trusted noise parameter $q$, with $\epsilon=10^{-10}$. The optimal value of $q$ and $\alpha$ are denoted $q^*$ and $\alpha^*$. Different lines refer to different values of the eavesdropper's disturbance: $p=0.10\,;\,0.11\,;\,0.1238$. The security parameter is fixed at $\epsilon=10^{-10}$. As the QBER increases, adding trusted noise becomes more beneficial, both in terms of absolute value of the secret-key rate value and in terms of minimum block length required for non-zero key generation.
• Figure 5: Maximum tolerable QBER plotted vs the sifted block length $m$. The two lines compare the QBER threshold, with optimal trusted noise and without trusted noise. The security parameter is fixed at $\epsilon=10^{-10}$. Up to $m=10^3$, trusted noise does not enhance the QBER tolerance. Trusted noise becomes beneficial for $m\ge10^4$. The asymptotic, known values, of $p_\text{max}=0.11$ and $p_\text{max}=0.124$ are retrieved for large block sizes.