Time coding protocols for quantum key distribution

We propose quantum key distribution protocols based on coherent single-photon optical pulses with duration $T$ and with minimum time-frequency uncertainty. The pulses are sent with possible delays (e.g., 0, $T∕2$) that are used to code the information (e.g., bit 0, bit 1) and that are shorter than the pulse width. Therefore, the time detection of the photons may result in a ambiguity of the delay evaluation for a potential eavesdropper. The duration of the received pulses is controlled thanks to a contrast measurement using an interferometer. A quantum formalism is given, allowing us to model the transmission of the key and the consequences of a possible eavesdropping. Two protocols are proposed and discussed. The first one involves two states and is limited to channels with losses lower than 50%. The second one involves four states, which prevents the eavesdropper from exploiting the losses of the line. The security of each protocol is evaluated as a function of channel losses, quantum bit error rate, and contrast loss in the case of intercept-resend attacks. It is applied to situations where photocounters dark counts are the main limitation of the system. The resulting maximum propagation distance allowing secure communication is evaluated.

Figure 1

Principle of the two-state protocol. Alice sends pulses of duration $T$ with chosen delay 0 or $T∕2$. Bob measures the photon detection time. The time slots 1 and 3 are nonambiguous and allow for delay determination. The time slot 2 is ambiguous and does not allow for delay determination.

Figure 2

Scheme of the experiment. Bob directs the pulses sent by Alice at random to a photon counter to establish the key or to a Mach-Zender interferometer that allows for duration measurement of the pulses.

Figure 3

Principle of the four-state protocol. Alice sends pulses of duration $T$ with chosen delays 0, $T∕2$, $T$, or $3T∕2$. Pulses (a) and (d) carry no information. Pulses (b) and (c) encode bit 0 and bit 1, respectively. Bob measures the photon detection time. He keeps only the results corresponding to time slot 3 and time slot 5. The results are ambiguous which prevents Eve from exploiting the losses of the line.

Figure 4

Information of Bob on Alice $\left(I_{AB}\right)$ and Eve on Alice for a relative contrast loss of 0% (a), 10% (b), and 20% (c) as a function of the quantum bit error rate $\left(Q\right)$. Eve sends pulses spanning only two time slots. For each curve, the maximum information of Eve on Alice is obtained when Eve intercepts all pulses (dashed line). For no loss of contrast, the maximum allowed value of $Q$ is 17%. For a relative contrast loss of 10%, the maximum allowed value of $Q$ is 9%.

Figure 5

Information of Bob on Alice $\left(I_{AB}\right)$ and Eve on Alice for a relative contrast loss of 0% (a), 10% (b), and 20% (c) as a function of the quantum bit error rate $\left(Q\right)$. Eve sends maximum coherence pulses (spanning three time slots). For each curve, the maximum information of Eve on Alice is obtained when Eve intercepts all pulses (dashed line). For no loss of contrast, the maximum allowed value of $Q$ is 5.8%. For a relative contrast loss of 10%, the maximum allowed value of $Q$ is 4.4%.