The BB84 QKD protocol is based on the idea that the sender and the receiver can reconcile a certain fraction of the teleported qubits to detect eavesdropping or noise and decode the rest to use as a private key. Under the present hardware infrastructure, decoherence of quantum states poses a significant challenge to performing perfect or efficient teleportation, meaning that a teleportation-based protocol must be run multiple times to observe success. Thus, performance analyses of such protocols usually consider the completion time, i.e., the time until success, rather than the duration of a single attempt. Moreover, due to decoherence, the success of an attempt is in general dependent on the duration of individual phases of that attempt, as quantum states must wait in memory while the success or failure of a generation phase is communicated to the relevant parties. In this work, we do a performance analysis of the completion time of the BB84 protocol in a setting where the sender and the receiver are connected via a single quantum repeater and the only quantum channel between them does not see any adversarial attack. Assuming certain distributional forms for the generation and communication phases of teleportation, we provide a method to compute the MGF of the completion time and subsequently derive an estimate of the CDF and a bound on the tail probability. This result helps us gauge the (tail) behaviour of the completion time in terms of the parameters characterising the elementary phases of teleportation, without having to run the protocol multiple times. We also provide an efficient simulation scheme to generate the completion time, which relies on expressing the completion time in terms of aggregated teleportation times. We numerically compare our approach with a full-scale simulation and observe good agreement between them.

翻译：BB84量子密钥分发协议基于这样一个理念：发送方和接收方可以协调一定比例的量子隐形传态量子比特来检测窃听或噪声，并解码其余部分以用作私钥。在现有硬件基础设施下，量子态的退相干对实现完美或高效的隐形传态构成了重大挑战，这意味着基于隐形传态的协议必须多次运行才能观察到成功。因此，此类协议的性能分析通常考虑完成时间，即成功所需的时间，而不是单次尝试的持续时间。此外，由于退相干，一次尝试的成功通常取决于该尝试各个阶段的持续时间，因为量子态必须在内存中等待，同时生成阶段的成功或失败信息会传达给相关方。在这项工作中，我们对发送方和接收方通过单个量子中继器连接且它们之间的唯一量子信道未受到任何恶意攻击的场景下，BB84协议的完成时间进行了性能分析。假设隐形传态的生成和通信阶段具有某种分布形式，我们提供了一种计算完成时间矩生成函数的方法，并随后推导出累积分布函数的估计值以及尾概率的边界。这一结果有助于我们根据描述隐形传态基本阶段的参数来评估完成时间的（尾部）行为，而无需多次运行协议。我们还提供了一种高效的仿真方案来生成完成时间，该方案依赖于将完成时间表示为聚合的隐形传态时间。我们将我们的方法与全规模仿真进行了数值比较，并观察到两者之间具有良好的吻合性。