Analytical bounds for decoy-state quantum key distribution with discrete phase randomization

TL;DR

Analytical bounds on the secret key generation rate of BB84 and measurement-device-independent QKD protocols in the DPR setting are derived and closely match the results obtained from more cumbersome numerical methods in the regions of interest.

Abstract

We analyze the performance of quantum key distribution (QKD) protocols that rely on discrete phase randomization (DPR). For many QKD protocols that rely on weak coherent pulses (WCPs), continuous phase randomization is assumed, which simplifies the security proofs for such protocols. However, it is challenging to achieve such a perfect phase randomization in practice. As an alternative, we can select a discrete set of global phase values for WCPs, but we need to redo the security analysis for such a source. While security proofs incorporating DPR have been established for several QKD protocols, they often rely on computationally intensive numerical optimizations. To address this issue, in this study, we derive analytical bounds on the secret key generation rate of BB84 and measurement-device-independent QKD protocols in the DPR setting. Our analytical bounds closely match the results obtained from more cumbersome numerical methods in the regions of interest.

Figures (8)

• Figure 1: Schematic diagram of the time-bin encoding BB84 protocol with DPR. Alice prepares a weak coherent pulse with a randomly chosen discrete phase $\theta_a\in\{2\pi j/D, j=0,..., D-1\}$. By choosing the polarization of the input pulse, we can use a Mach-Zehnder interferometer to encode the input in $Z$ and $X$ bases. The resulting pulses are transmitted through a quantum channel to Bob. Upon receiving the pulses, Bob randomly selects his measurement basis. For the $Z$ basis, he determines the logical bit by recording whether the detection event occurs in the early or late time bin. For the $X$ basis, Bob measures the phase difference between r and s pulses. PBS: Polarizing beam splitter; PM: Phase modulator; OS: Optical switch; BS: Beam splitter; HWP: Half-wave plate.
• Figure 2: Schematic diagram of the time-bin encoding MDI QKD protocol with DPR. Both Alice and Bob independently prepare weak coherent pulses with randomly chosen discrete phase values $\theta_a ,\theta_b\in\{2\pi j/D,j=0,...,D-1\}$ with respect to their local phase references. Each prepared state is passed through a BB84 encoder as in \ref{['fig:BB84 Scheme']}. The resulting optical pulses are then transmitted through quantum channels to an untrustworthy intermediary, Charlie, who performs an interference measurement on the incoming states. Charlie publicly announces the detection results. Based on this announcement and their own encoding choices, Alice and Bob can post-select correlated events to extract a shared key bit.
• Figure 3: Secret key generation rates for the DPR BB84 protocol using numerical (solid lines) and analytical (dashed) methods. The mean number of photons for the signal and decoy states are optimized to get the maximum key rate. The best performing curve is based on the vacuum+weak decoy states and CPR. The remaining curves are generated when considering $D$ phase slices and vacuum+weak decoy states.
• Figure 4: The upper bound $e_{x,1}^{b,\mu U}$ on the bit error rate for the DPR BB84 protocol using numerical (solid) and analytical (dashed) methods. Results for the CPR case are shown as well.
• Figure 5: The optimized value of $\mu$ for different number of phase slices ranging from $D=5$ to $D=10$, versus distance in the DPR BB84 protocol. A CPR related curve is provided for comparison. The numerical results are given in solid lines while analytical results are given in dashed lines.