Security proof of practical quantum key distribution with detection-efficiency mismatch

Quantum key distribution (QKD) protocols with threshold detectors are driving high-performance QKD demonstrations. The corresponding security proofs usually assume that all physical detectors have the same detection efficiency. However, the efficiencies of the detectors used in practice might show a mismatch depending on the manufacturing and setup of these detectors. A mismatch can also be induced as the different spatial-temporal modes of an incoming signal might couple differently to a detector. Here we develop a method that allows to provide security proofs without the usual assumption. Our method can take the detection-efficiency mismatch into account without having to restrict the attack strategy of the adversary. Especially, we do not rely on any photon-number cutoff of incoming signals such that our security proof is directly applicable to practical situations. We illustrate our method for a receiver that is designed for polarization encoding and is sensitive to a number of spatial-temporal modes. In our detector model, the absence of quantum interference between any pair of spatial-temporal modes is assumed. For a QKD protocol with this detector model, we can perform a security proof with characterized efficiency mismatch and without photon-number cutoff assumption. Our method also shows that in the absence of efficiency mismatch in our detector model, the key rate increases if the loss due to detection inefficiency is assumed to be outside of the adversary's control, as compared to the view where for a security proof this loss is attributed to the action of the adversary.

Figure 1

Schematic of Bob's measurement device: (a) and (b) describe the active-detection and passive-detection schemes, respectively. To actively or passively select a measurement basis, a polarization rotator or a 50/50 beam splitter is used. Under each basis, a polarizing beam splitter and two detectors are used to measure the polarization state of an incoming optical signal. Each detector is labeled by the associated measurement outcome.

Figure 2

Constructive description of the squashing map $\mathrm{\Lambda }$ for the flag-state squasher. Each line corresponds to a subspace of the input Hilbert space associated with the block-diagonal decomposition of the POVM elements $M_y$ with $y\in \{1,\cdots ,J\}$, as indicated on the left side.

Figure 3

The minimum double-click probability $d_{n,\min }$ vs. the photon number $n$ received by Bob for the active-detection mismatch model of Table 1 with ${\eta }_1=1$ and ${\eta }_2=\eta$. Note the monotonicity of each curve as a function of $n$ and that $d_{2,\min }$, as well as $d_{1,\min }$, is always equal to zero.

Figure 4

The minimum effective-error probability $e_{n,\min }$ vs the photon number $n$ received by Bob for the active-detection mismatch model of Table 1 with ${\eta }_1=1$ and ${\eta }_2=\eta$. Note that $e_{3,\min }$ is a lower bound on $e_{n,\min }$ when $n\ge 3$ and that $e_{1,\min }$ is always equal to zero.

Figure 5

The minimum cross-click probability $c_{n,\min }$ vs. the photon number $n$ received by Bob for the passive-detection mismatch model of Table 2 with ${\eta }_1=1$ and ${\eta }_2=\eta$. Note the monotonicity of each curve as a function of $n$, supporting the inequalities in Eqs. (24) and (25).

Figure 6

Reliable key-rate lower bound in bits per round obtained by our numerical method vs the detection efficiency $\eta$ of all detectors used in Bob's measurement device as shown in Fig. 1. We consider both the active- and passive-detection schemes. For data simulation, we fix the depolarizing probability $\omega =0.05$, the multi-photon probability $r=0.05$, and the product of transmission efficiency $t$ and detection efficiency $\eta$ to be $t\eta =0.1$. We choose these values just for ease of graphical illustrations. We remark that under each detection scheme, the probability distribution observed by Alice and Bob does not change with $\eta$ as long as the simulation parameters $\omega$, $r$, and $t\eta$ are fixed.

Figure 7

Reliable key-rate lower bound in bits per round obtained by our numerical method vs. the detection efficiency ${\eta }_2$ of the detector labeled by “$V/A$” (for the signals stayed in the first spatial-temporal mode) in the active-detection scheme of Fig. 1. For data simulation, we fix the detection efficiency of the detector labeled by “$H/D$” (for the signals stayed in the first spatial-temporal mode) to ${\eta }_1=0.2$. We also fix the depolarizing probability $\omega =0.05$, the multi-photon probability $r=0.05$, and the transmission efficiency $t=0.5$ (corresponding to $3\mathrm{dB}$ loss). We remark that for the active-detection scheme the key rate scales linearly with the probability $p$ for Bob to select the key-generation basis when other simulation parameters are fixed, and that for the results shown in this figure and the following Fig. 8 the probability $p$ is fixed to be $1/2$ according to the data-simulation model detailed in Sec. 4a.

Figure 8

Reliable key-rate lower bound in bits per round obtained by our numerical method vs the transmission distance in kilometers from Alice to Bob with the active-detection scheme of Fig. 1. For data simulation, we fix the detection efficiencies of the two detectors labeled by “$H/D$” and by “$V/A$” (for the signals stayed in the first spatial-temporal mode) to ${\eta }_1=0.2$ and ${\eta }_2=0.18$, respectively. We also fix the depolarizing probability $\omega =0.05$ and the multiphoton probability $r=0.05$. Note that when the same photon-number flag threshold $k=2$ is used, the key-rate lower bound obtained for the case of mode-dependent efficiency mismatch is slightly lower than that for the case of mode-independent efficiency mismatch, although due to the use of a logarithmic scale in the plot such a difference is hard to be visible.

Figure 9

Reliable key-rate lower bound in bits per round obtained by our numerical method vs. the detection efficiency ${\eta }_2$ of the detectors labeled by “$V$,” “$D$,” or “$A$” (for the signals stayed in the first spatial-temporal mode) in the passive-detection scheme of Fig. 1. For data simulation, we fix the detection efficiency of the detector labeled by “$H$” (for the signals stayed in the first spatial-temporal mode) to ${\eta }_1=0.2$. We also fix the depolarizing probability $\omega =0.05$, the multiphoton probability $r=0.05$, and the transmission efficiency $t=0.5$ (corresponding to $3\mathrm{dB}$ loss).

Figure 10

Reliable key-rate lower bound in bits per round obtained by our numerical method vs the transmission distance in kilometers from Alice to Bob with the passive-detection scheme of Fig. 1. For data simulation, we fix the detection efficiencies ${\eta }_1=0.2$ and ${\eta }_2=0.18$, where ${\eta }_1$ and ${\eta }_2$ have the same meanings as those in Fig. 9. We also fix the depolarizing probability $\omega =0.05$ and the multi-photon probability $r=0.05$. Note that when the same photon-number flag threshold $k=1$ is used, the key-rate lower bound obtained for the case of mode-dependent efficiency mismatch is slightly lower than that for the case of mode-independent efficiency mismatch. Also, by comparing the results obtained using the photon-number flag threshold $k=1$ with those obtained using $k=2$ in the case of one spatial-temporal mode, we can see that the usage of the photon-number flag threshold $k=1$ can induce a nonmonotonic behavior of the obtained key-rate lower bound as a function of the transmission distance.