Securing quantum key distribution systems using fewer states

Quantum key distribution (QKD) allows two remote users to establish a secret key in the presence of an eavesdropper. The users share quantum states prepared in two mutually unbiased bases: one to generate the key while the other monitors the presence of the eavesdropper. Here, we show that a general $d$-dimension QKD system can be secured by transmitting only a subset of the monitoring states. In particular, we find that there is no loss in the secure key rate when dropping one of the monitoring states. Furthermore, it is possible to use only a single monitoring state if the quantum bit error rates are low enough. We apply our formalism to an experimental $d=4$ time-phase QKD system, where only one monitoring state is transmitted, and obtain a secret key rate of $17.4\pm 2.8$ Mbits/s at a 4 dB channel loss and with a quantum bit error rate of $0.045\pm 0.001$ and $0.037\pm 0.001$ in time and phase bases, respectively, which is 58.4% of the secret key rate that can be achieved with the full setup. This ratio can be increased, potentially up to 100%, if the error rates in time and phase basis are reduced. Our results demonstrate that it is possible to substantially simplify the design of high-dimensional QKD systems, including those that use the spatial or temporal degrees of freedom of the photon, and still outperform qubit-based ($d=2$) protocols.

Figure 1

The secret key fraction (a) and the numerically obtained upper bound on the phase error rate (b) plotted as a function of the quantum bit error rate for $d=4$.

Figure 2

The secret key fraction (a) and the upper bound on the phase error rate (b) plotted as a function of the quantum bit error rate when Alice transmits $d-1$ states and only one monitoring-basis state.

Figure 3

(a) Illustration of the four dimensional time-phase states. (b) Experimental setup of $d=4$ time-phase QKD.

Figure 4

(a) The error rates corresponding to each of the phase-basis states plotted as a function of the channel loss. (b) Asymptotic secret key rates for the cases where Alice transmits one or four phase-basis states are plotted as a function of quantum channel loss.

Figure 5

Dependence of secret key rate on the quantum channel loss when (a) the detectors are assumed to have a fixed efficiency of 75% and (b) the detectors are assumed to have a rate-dependent efficiency.