High-dimensional measurement-device-independent quantum key distribution on two-dimensional subspaces

Quantum key distribution (QKD) provides ultimate cryptographic security based on the laws of quantum mechanics. For point-to-point QKD protocols, the security of the generated key is compromised by detector side channel attacks. This problem can be solved with measurement-device-independent QKD (mdi-QKD). However, mdi-QKD has shown limited performances in terms of the secret key generation rate, due to postselection in the Bell measurements. We show that high-dimensional (Hi-D) encoding (qudits) improves the performance of current mdi-QKD implementations. The scheme is proven to be unconditionally secure even for weak coherent pulses with decoy states, while the secret key rate is derived in the single-photon case. Our analysis includes phase errors, imperfect sources, and dark counts to mimic real systems. Compared to the standard bidimensional case, we show an improvement in the key generation rate.

Figure 1

Schematic of the proposed setup for Hi-D-mdi QKD. (a) Space is used to encode information in different paths (multicore fibers can be used as transmission channels). $2N$ single-photon detectors are necessary for this configuration. (b) Time-encoding scheme, where different time slots are used to encode the qudits. The number of detectors is independent of the dimension $N$.

Figure 2

Secure key rate as a function of distance. Plain lines refer to $N=2$, dash-dotted lines to $N=3$, dashed lines to $N=4$, and dotted lines to $N=8$. (a,b) No detector dead time, ${\tau }_d=0$. The secret key rate without detector dead time $\tilde{r}$ is found using Eq. (2), with $R$ substituted by $R_p$, i.e., $\tilde{r}$ is in $bit$ per application of the protocol. (c) Secret key rate per second $r$ as a function of distance. The dead time is ${\tau }_d=20$ ns, and the minimum pulse separation is ${\tilde{T}}_p=200$ ps $\left({\tau }_d/{\tilde{T}}_p=100\right)$. Common parameters are $P_{\mathrm{dc}}=1\times {10}^{-6},f\left({\epsilon }_z\right)=1,{\left|\beta \right|}^2=0.85,\eta =0.145$, and $\sigma$ equal to 0.175 (time) or 0.325 (space). $\sigma$ is chosen such that, for $N=2$ when only including dephasing, there is a QBER ${\epsilon }_x$ of $1.5\%$ (time) or $5\%$ (space).

Figure 3

Raw key per detector $R_{\text{det}}=R/n_{\det }$ as a function of the dimension $N$, in the detector saturation regime. Violet circles (solid and empty) are used for the space encoding. Blue squares (solid and empty) are used for the time encoding. The number of pulses $n$ within ${\tau }_d$ is optimized to achieve the highest rate. The maximum possible number of qubits ${\tau }_d/{\tilde{T}}_p$ is equal to either 20 (empty circles and squares) or 100 (solid circles and squares). $P_s=0.2,{\tau }_d=20$ ns, and $n_{\det }=2N$ (space) or $n_{\det }=2$ (time).