# Qubit-Based Clock Synchronization for QKD Systems Using a Bayesian Approach
## 1 Introduction
In this way, small differences in clock frequencies can gradually change the clock offset so that a previously calculated synchronization is no longer valid. Other timing errors, such as clock jitter and frequency drift, also contribute to the need for a more robust synchronization solution. We denote the time over which synchronization is maintained as \(T_{b}\), i.e., the time over which the error in \(\Delta\ll\tau_{A}\). Clock synchronization is sometimes achieved by directly sending [PERSON]'s clock signal to [PERSON] over a separate channel via an optical link or using a radio-frequency signal [7; 8; 9; 10; 11; 12; 13; 14; 15; 16]. However, this introduces additional hardware requirements and increases the cost and complexity of the setup. One way to avoid these additional resource requirements is to use the quantum channel itself to transmit the information necessary to perform the synchronization [17; 18; 19; 20]. One such qubit-based synchronization protocol was introduced and demonstrated by [PERSON] et al. [17]. Their protocol uses a dedicated clock-synchronization phase followed by a key distribution phase. In the synchronization phase, a pre-agreed synchronization string is transmitted to [PERSON] and the clocks are aligned during post-processing. The pre-agreed synchronization string is used to find the initial offset between [PERSON] and [PERSON]'s clocks. Because it must be public knowledge, it cannot be used to generate a secure key. If the clock frequencies are not consistent, simple clock offset recovery only temporarily aligns until the clock drift becomes of-the-order-of the communication protocol temporal bin width \(\tau_{A}\). Correcting for this clock frequency drift using only clock offset recovery requires repeated synchronization/key distribution phases with a regularity that depends on the stability of the clocks used in the experiment. This reduces the overall secure key rate because no QKD states can be sent while the synchronization states are being sent, which may result in zero key rate due to finite-key effects [21; 22]. However, to account for this drift without needing to send regular synchronization strings, [PERSON] et al. performs clock frequency recovery using the periodic arrival times of [PERSON]'s qubits. Unlikethe clock offset recovery, this does not require a pre-agreed synchronization string and thus does not decrease the amount of key that can be sent. While our method only uses clock offset recovery at this time, it avoids these potential impacts on secure key rate by synchronizing the clocks using only information that is already publicly sent over the insecure classical channel by [PERSON] and [PERSON] for sifting and security analysis: The basis choices and the mean photon number of the transmitted signal. Because we are transmitting only one state in the monitoring basis, the basis choices provide information about which of [PERSON]'s measurement outcomes are more likely. The decoy state choices, which determine [PERSON]'s mean photon number for each wavepacket, also contain information about [PERSON]'s measurement outcomes. For example, [PERSON] is unlikely to record any detections if [PERSON] sends the vacuum decoy state. By comparing this information to his measurement outcomes, [PERSON] can probabilistically determine the timing offset. To account for potential clock drift, [PERSON] can perform this synchronization in subsets of length \(T_{b}\). Thus, [PERSON] can find the up-to-date timing offset and ensure that the basis choices he publishes are properly lined up with the ones sent to him by [PERSON], but this requires an efficient analysis method to reduce the data requirements. Of course, our approach as well as [PERSON]'s requires low enough channel loss so that there are enough events received by [PERSON] over a drift interval as discussed below. Another example of a qubit-based synchronization protocol for continuously-pumped entanglement-based QKD systems was introduced by [PERSON] et al. [20].