Effect of detector dead times on the security evaluation of differential-phase-shift quantum key distribution against sequential attacks

We investigate limitations imposed by detector dead times on the performance of sequential attacks against a differential-phase-shift (DPS) quantum key distribution (QKD) protocol with weak coherent pulses. In particular, we analyze sequential attacks based on unambiguous state discrimination of the signal states emitted by the source and we obtain ultimate upper bounds on the maximal distance achievable by a DPS QKD scheme both for the case of calibrated and uncalibrated devices, respectively.

Figure 1

Basic setup of a DPS QKD scheme. PM denotes a phase modulator, BS, a 50:50 beam splitter, $M$, a mirror, $D0$ and $D1$ are two photon detectors and $\Delta t$ represents the time difference between two consecutive pulses.

Figure 2

Possible signal states that Eve sends to Bob together with their a priori probabilities. The arrow indicates the transmission direction.

Figure 3

Labeling convention for the $k$ temporal modes of the signal state $|{\psi }_k⟩$ given by Eq. (4) followed by $1+d$ vacuum states. The arrow indicates the transmission direction.

Figure 4

Gain $\left(G\right)$ versus QBER in a sequential attack for three different distributions of the state coefficients $A_n^{\left(k\right)}$: Flat (solid line), binomial (dashed line), and the optimal distribution (dotted line). The mean photon number of Alice’s signal states is ${\mu }_{\alpha }=0.2$, and the parameter $d=500$. The triangles represent experimental data from Ref. [21].

Figure 5

Gain $\left(G\right)$ versus QBER in a sequential attack for three different distributions of the state coefficients $A_n^{\left(k\right)}$: Flat (solid line), binomial (dashed line), and the optimal distribution (dotted line). The mean photon number of Alice’s signal states is ${\mu }_{\alpha }=0.17$, and the parameter $d=50$. The triangles represent experimental data from Ref. [18]. (See also Ref. [20].)

Figure 6

Gain $\left(G\right)$ versus QBER in a sequential attack for three different distributions of the state coefficients $A_n^{\left(k\right)}$: Flat (solid line), binomial (dashed line), and the optimal distribution (dotted line). The mean photon number of Alice’s signal states is ${\mu }_{\alpha }=0.16$, and the parameter $d=50$. The triangle represents experimental data from Ref. [18]. (See also Ref. [20].)

Figure 7

Gain $\left(G\right)$ versus QBER in a sequential attack for three different distributions of the state coefficients $A_n^{\left(k\right)}$: Flat (solid line), binomial (dashed line), and the optimal distribution (dotted line). The mean photon number of Alice’s signal states is ${\mu }_{\alpha }=0.2$, and the parameter $d=50$. The triangles represent experimental data from Ref. [19]. (See also Ref. [20].)

Figure 8

Possible signal states arriving at Bob’s detection apparatus after a dead time, together with their a priori probabilities. The arrow indicates the transmission direction.

Figure 9

Gain $\left(G\right)$ versus QBER in a sequential attack for different values of the photon number $m$, and for the optimal distribution of the state coefficients $A_{n,m}^{\left(k\right)}$ derived in Sec. 4. The mean photon number of Alice’s signal states is ${\mu }_{\alpha }=0.2$, the parameter $\tilde{d}=500$, the dark count probability of Bob’s detectors is $p_d=2.5\times {10}^{-9}$, and the detection efficiency ${\eta }_{\text{det}}=0.005$. The triangles represent experimental data from Ref. [21].

Figure 10

Gain $\left(G\right)$ versus QBER in a sequential attack for different values of the photon number $m$, and for the optimal distribution of the state coefficients $A_{n,m}^{\left(k\right)}$ derived in Sec. 4. The mean photon number of Alice’s signal states is ${\mu }_{\alpha }=0.17$, the parameter $\tilde{d}=50$, the dark count probability of Bob’s detectors is $p_d=7.8\times {10}^{-6}$, and the detection efficiency ${\eta }_{\text{det}}=0.0327$. The triangle represents experimental data from Ref. [18]. (See also Ref. [20].)

Figure 11

Gain $\left(G\right)$ versus QBER in a sequential attack for different values of the photon number $m$, and for the optimal distribution of the state coefficients $A_{n,m}^{\left(k\right)}$ derived in Sec. 4. The mean photon number of Alice’s signal states is ${\mu }_{\alpha }=0.16$, the parameter $\tilde{d}=50$, the dark count probability of Bob’s detectors is $p_d=2.7\times {10}^{-7}$, and the detection efficiency ${\eta }_{\text{det}}=0.0045$. The triangle represents experimental data from Ref. [18]. (See also Ref. [20].)

Figure 12

Gain $\left(G\right)$ versus QBER in a sequential attack for different values of the photon number $m$, and for the optimal distribution of the state coefficients $A_{n,m}^{\left(k\right)}$ derived in Sec. 4. The mean photon number of Alice’s signal states is ${\mu }_{\alpha }=0.2$, the parameter $\tilde{d}=50$, the dark count probability of Bob’s detectors is $p_d=3.5\times {10}^{-8}$, and the detection efficiency ${\eta }_{\text{det}}=0.0011$. The triangles represent experimental data from Ref. [19]. (See also Ref. [20].)

Figure 13

Gain $\left(G\right)$ versus double click rate $\left(D_c\right)$ in a sequential attack for different values of the photon number $m$, and for the optimal distribution of the state coefficients $A_{n,m}^{\left(k\right)}$ derived in Sec. 6. The experimental parameters coincide with those used in Fig. 10.

Figure 14

The average total number of errors $e\left(k\right)$ versus $k$ for different distributions of the state coefficients $A_n^{\left(k\right)}$: Flat (solid line), binomial (dashed line), and the optimal distribution (dotted line).