# In-Field Quantum-Protected Control-Based Key Distribution with a Lossy Urban Fiber Link
## 5 Results  
Figure 5 shows the experimentally observed dynamics of the interceptable leakage \(r_{\rm E}\) and secret key rate \(R\). The horizontal axis corresponds to the number of conducted cycles of Transmission and Control (Steps 1-3 of the protocol), \(N\). Within the time interval reflected in Figure 5, the measured \(r_{\rm E}\) varies between 19% and 22%, while \(R\) correspondingly changes within the range from \(2\times 10^{-4}\) and \(2.7\times 10^{-3}\) bits per channel use. The average \(R\) is \(8\times 10^{-4}\) bits per channel use, which translates into 490 bit/s given that the average rate at which Alice transmits bits is 614 kbit/s. Figure 4: Secret key rate \(R\) (in bits per channel use) as a function of interceptable leakage \(r_{\rm E}\) for different advantage distillation block lengths \(M\). The white region indicates the range of \(r_{\rm E}\) observed in the experiment. Figure 3: Critical \(r_{\rm E}\) providing positive secret key generation rate for \(M=2\) depicted as a function of signal intensities. Light yellow parts of the graphs correspond to the pairs of intensities \((\mu_{0},\mu_{1})\) for which every \(r_{\rm E}\) is acceptable, as long as the legitimate users can achieve \(\bar{p}_{\rm err}\leq 5\%\) after advantage distillation. **(a)** The signal intensities are sampled from sets with relatively wide boundaries: \(2270\leq\mu_{0}\leq 2770\), \(2320\leq\mu_{1}\leq 2820\). **(b)** The signal intensities are sampled from sets with more narrow boundaries: \(2320\leq\mu_{0}\leq 2420\), \(2470\leq\mu_{1}\leq 2570\). Due to the intrinsic transmittometry error, on a short scale of \(N\), the measured value of \(r_{\rm E}\) fluctuates by \(\delta r_{\rm E}\) (this deviation is about \(\delta r_{\rm E}\approx 0.15r_{\rm E}\)). With that, the selected measurement time interval in Figure 5 is such that any systematic error due to the system's detuning (see Section 7.1 in Methods for details) remains below the same threshold of \(\delta r_{\rm E}\). When calculating the compression factor for the privacy amplification, \(r_{\rm E}\) fluctuations are taken into account as \(r_{\rm E}+3\delta r_{\rm E}\). Equation (1) relates the key rate and Eve's information \(I_{\rm M}(A,E)\). And this quantity, \(I_{\rm M}(A,E)\), fluctuates due to \(r_{\rm E}\) fluctuations \(\delta r_{\rm E}\). Figure 4 illustrates how the key rate depends on \(r_{\rm E}\), e.g., for \(M=2\), \(\delta R/\delta r_{\rm E}\approx-4.6\times 10^{-2}\). Note that the secret key rate \(R\) in Figure 5 appears constant over intervals of \(\tilde{N}=3\) to 10 Transmission and Control cycles. This is because the used error correction implementation required users to have a minimum number of raw bits, which cannot be harvested within a single cycle. Consequently, secret keys are generated only after sufficient bits have been accumulated, typically over \(\tilde{N}=3\) to 10 cycles, depending on the error rate. Although each cycle has its own measured \(r_{\rm E}\), in privacy amplification of the combined data, we apply the maximum from the measured \(r_{\rm E}\) values to the entire accumulated sequence. Within each \(\tilde{N}\) interval, the plotted \(R\) represents an average over those cycles, which looks like a constant. The statistical quality of the generated key is demonstrated in Figure 6, where we plot in Figure 6a the mean and Figure 6b the variance in the consecutive \(n\) bits of the key as function of \(n\), and compare these values to those expected from an ideal equiprobable binomial distribution. Analyzing statistics within consecutive blocks of length \(n\) is beneficial since it fully reflects the quality of the key in respect to the potential noises of different  
Figure 5: Dynamics of the secret key rate and losses. The lower blue trace shows the secret key rate \(R\) (in bits per channel use) as a function of the number of Transmission and Control cycles \(N\). The upper red trace displays the corresponding value of \((1-r_{\rm E})\).