Abstract

Security certification of quantum key distribution systems with a practical device is essential for their social deployment. Considering the transmitter, we investigate quantum state generation affected by degraded electrical signals from practical bandwidth-limited devices on high-speed phase-encoding BB84 quantum key distribution systems. The state preparation flaw caused by this degradation undesirably enhances the distinguishability between the two bases for the BB84 protocol and decreases the key generation rate. We propose the state preparation with a dual parallel modulator for increasing the robustness to signal degradation. To verify the effectiveness of the dual parallel modulator, we characterize the generated states using state tomography and estimate the key generation rate based on the Gottesman–Lo–Lütkenhaus–Preskill theory with fidelity derived from the estimated density matrices. Simulation results show that the key generation rate remains unaffected by modulation voltage shifts up to 20%.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Since its proposal in 1984 [1], quantum key distribution (QKD) has been extensively developed in theory and implementation [2]. QKD allows two distant parties, Alice and Bob, to share secure cryptographic keys, even in the presence of an eavesdropper, Eve. The performance of QKD systems has improved in recent years. For example, ultrafast QKD systems operating with above-gigahertz clock speeds provide secure key distribution at several hundred kilobits per second over 50 km transmissions [36]. Moreover, systems installed on fiber networks can continuously operate for months [712]. Despite such remarkable achievements, QKD systems have not been widely deployed in mainstream networks. One of the main obstacles for QKD adoption is the lack of security certifications for practical devices. Although the unconditional security of the BB84 protocol for QKD has been proved theoretically [13], the proof requires several conditions to be satisfied. Among them, the transmitter should encode the information on the photon states without introducing errors, but this premise may not be fully satisfied in practice due to device imperfections.

Most high-speed QKD systems implementing the phase-encoding BB84 protocol use modulators based on lithium niobate for phase and intensity modulation controlled by electric signals during state preparation. Although the lithium niobate modulators are suitable for conventional optical communication systems, high-speed QKD systems can be adversely affected by imperfect modulation [14]. In fact, the state preparation flaw [15], that is, the difference between the modulated and ideal states, arises from degraded electrical signals applied to the modulators. Although signal degradation due to practical bandwidth-limited devices is common in high-speed systems, it substantially increases the error rate in QKD systems compared to conventional light-wave communication systems, given the analog nature of quantum states [16]. Moreover, discrepancy between prepared quantum states and theoretical quantum states caused by imperfection of device, especially the phase modulator which will be discussed in this article, severely decreases the key generation rate given the increased channel loss [17]. Therefore, modulation devices should be more tolerant to degraded electric signals for realizing security certified ultrafast QKD systems. Although a loss-robust protocol is recently available [18], it requires to constrain the state errors into a small range to provide high key generation rate.

In this paper, we propose using a dual parallel modulator (DPM) to mitigate the state preparation flaw. We show that the DPM allows to generate QKD states that are robust to inaccurate voltage signals. In Section 2, we quantitatively define the state preparation flaw with fidelity between conjugated-basis states to relate the corresponding density matrices. Then, we examine the mechanism of state preparation flaw in conventional modulators and introduce the proposed DPM approach. In Section 3, we describe the method and results of state characterization using state tomography. In Section 4, we estimate the state preparation flaw caused by varying the supply voltage based on the results of state tomography. We also calculate the secure key generation rate and show that this rate is not affected by degraded modulation signals in the proposed approach. We finally draw conclusions in Section 5.

2. State generation for phase-encoding BB84 QKD and state preparation flaw

As shown in Fig. 1, in a phase-encoding BB84 QKD system, Alice sends a photon qubit in state ${|\psi }\rangle = [{|{0\rangle + \textrm{exp}({i\theta } )} |1\rangle } ]/\sqrt 2 $ to Bob. The time-bin qubit basis $\{ |0\rangle, |1\rangle\} $ is generated from an optical pulse by passing a planar light-wave circuit asymmetric Mach–Zehnder interferometer (PLC-AMZI), where $\textrm{|0}\rangle $ and $\textrm{|1}\rangle $ are defined as the former and latter pulse amplitudes, respectively. Relative phase θ is randomly selected by a phase modulator from set {0, π/2, π, 3π/2} to define the BB84 states {X0, X1} = {0, π} and {Y0, Y1} = {π/2, 3π/2} for bit values 0 and 1 in the X and Y bases, respectively. Bob measures the quantum state, after randomly selecting a measurement basis by adding relative phase of 0 or π/2. After transmission, Bob informs Alice of his basis selection. If the bases chosen by Alice and Bob in a pulse differ, they discard the measurement.

 figure: Fig. 1.

Fig. 1. Typical implementation of phase-encoding BB84 QKD system (LD, laser diode; AMZI, asymmetric Mach–Zehnder interferometer; PM, phase modulator; ATT, attenuator; PD, photon detector). The inset shows a deformed square wave observed in a high-speed QKD system. A 20% fluctuation occurs in point A and B in square wave.

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Under this setup, we place a phase modulator after the PLC-AMZI given the current unavailability of monolithic devices integrating an PLC-AMZI and a phase modulator. PLC-AMZI takes the advantages over fiber based AMZI, because it provides stable interference without complicated dynamic control of time delay [1921]. However, the modulation frequency is doubled to generate quantum states, the modulator must operate at a speed as twice the clock frequency. The limited bandwidth of the drive circuit affects more severely than that used for fiber based AMZI. In recent high-speed QKD systems, the system clock reaches 1 GHz. With such high frequency, we observe that electric signal to the modulator varies 20% in 200 ps from time A to B as illustrated in the insert of Fig. 1 which is longer than the pulse duration of light (100 ps). Therefore, the phase modulation to the light pulse differs up to 20%, if the timing of light pulse fluctuates from time A to B. This inaccuracy can produce errors in the relative phase and consequently generate the state preparation flaw. Although limited-bandwidth devices have been investigated on light intensities regarding the decoy effect [16], it also affects the accuracy of state preparation.

The errors in state preparation not only increase the bit error rate but also affect the security of QKD. Ideally, completely mixed states in the X basis, ${{\rho}_\textrm{X}}$, and Y basis, ${{\rho}_\textrm{Y}}$, are equal to half the identity matrix, thus being indistinguishable. However, the state preparation flaw may enhance the distinguishability between the two-basis states when implementing the BB84 protocol. The state preparation flaw $\Delta$ is quantitatively defined in terms of the fidelity F between states as

$$\Delta = \frac{{1 - {\mathop{F}\nolimits} ({{\rho_{\rm X}},{\rho_{\rm Y}}} )}}{2},$$
$${\mathop{F}\nolimits} ({{\rho_{\rm X}},{\rho_{\rm Y}}} ) = \textrm{tr}\left( {\sqrt {{{\left( {\sqrt {{\rho_{\rm X}}} \sqrt {{\rho_{\rm Y}}} } \right)}^\dagger }\sqrt {{\rho_{\rm X}}} \sqrt {{\rho_{\rm Y}}} } } \right).$$
The increased distinguishability between states implies that Eve can obtain more information than that estimated from the phase error rate. Hence, the state preparation flaw increases the effective phase error rate from ${\delta _\textrm{Y}}$ estimated from the observed error rate [18]:
$${\delta ^{\prime}_{\rm Y}} = {\delta _{\rm Y}} + 4\Delta + 4\sqrt {\Delta {\delta _{\rm Y}}}.$$
Consequently, the final key generation rate, R, decreases by replacing the phase error rate in the formula given with the decoy state method [2325] in the asymptotic limit by
$$R \ge q[{{{Q}_1}({1 - H({{\delta^{\prime}_{\rm Y}} })} )- {Q_{\rm s}}fH({\delta_{\rm X}})} ],$$
where ${\delta _\textrm{X}}$ is the bit-flip error rate, whereas ${Q_\textrm{1}}$ and ${Q_\textrm{s}}$ represent the gains of the single-photon states sent by Alice in the received pulses and the signal states, respectively. We here assume key distillation from X-basis pulses and phase error estimation using Y-basis pulses. A constant q is given by the selection ratio of the X basis [26]. The key generation rate relates to the bit and phase error rates through Shannon binary entropies $H({{\delta_\textrm{X}}} )$ and $H({\delta {^{\prime}_\textrm{Y}}} )$ in the signal pulses. Parameter f represents the efficiency of error correction.

In high-speed QKD systems, the dual-drive modulator (DDM) illustrated in Fig. 2 is often used to apply phase differences [22]. DDM is based on Mach–Zehnder interferometer (MZI) and it modulates light intensity and phase by applying phase shifts independently on upper and lower paths in Fig. 2. The phase shifts ${\phi _\textrm{1}}$ and ${\phi _\textrm{2}}$ in the upper and lower paths are induced by applied voltages ${\textrm{V}_{{\phi _\textrm{1}}}}$ and ${{V}_{{\phi _\textrm{2}}}}$ to the upper and lower electrodes, respectively. Then, the electric field of input light ${{E}_\textrm{i}}$ is transformed into that of output light ${{E}_\textrm{o}}$ as follows:

$${E_o} = \frac{1}{2}({{e^{i{\phi_1}}} + {e^{i{\phi_2}}}} )= \cos \frac{{{\phi _1} - {\phi _2}}}{2}{e^{i\frac{{{\phi _1} + {\phi _2}}}{2}}}{E_i}.$$
Then the photon state is given by:
$$|\Psi \rangle=\frac{1}{{\sqrt 2 }}\sum\limits_{j = 0,1} {\exp (} i\frac{{\phi _1^j + \phi _2^j}}{2})\cos \frac{{{\phi _1}^j - {\phi _2}^j}}{2}|j \rangle .$$

 figure: Fig. 2.

Fig. 2. Schematic of DDM (${{E}_\textrm{i}}$, electric field of input light; ${{V}_\phi }$, applied voltage to electrodes; $\phi $, phase shift; ${{E}_{o}}$, electric field of output light). DDM is based on MZI and modulates light intensity and phase by applying phase shifts independently on upper and lower paths.

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By appropriately setting voltages ${{V}_{{\phi _{1}}}}$ and ${{V}_{{\phi _{2}}}}$, we can generate the four states required for the phase-encoding BB84 protocol, ${|\psi }\rangle = [{|{0\rangle + \textrm{exp}({i\theta } )} |1\rangle } ]/\sqrt 2, \; \theta \in \{{0, {\pi }/2, {\pi}, \;3 {\pi }/2} \}$, as shown in Eq. (6) and Table 1 [22]. Here global phases are ignored. DDMs are suitable to implement the BB84 protocol, because the combinations of only two phase values on each electrode can generate the four states. Moreover, the difference between the applied voltages is ${{V}_{\pi}}$, as listed in Table 1. Thus, the design of the driving circuit is considerably simplified.

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Table 1. Phase shifts in DDM to generate photon states X0, X1, Y0, and Y1. Variable $\phi _{i}^{({j} )}$ (i = 1, 2, j = 0, 1) denotes phase applied to i-th electrode for state $|{j}\rangle$.

However, the DDM is sensitive to inaccuracies of the applied voltage, because the error in its phase modulation is proportional to the voltage shift. Suppose a voltage shift ${{V}_\varepsilon }$ occurs on ${{V}_{{\phi _\textrm{2}}}}$ (the electrode on the lower branch). As the electro-optical effect is linear with the applied electric field, phase shift error ɛ to ${\phi _\textrm{2}}$ also varies linearly according to voltage shift ${{V}_\varepsilon }$. Then, the coefficients in Eq. (6) change to

$$\begin{array}{c} {\textrm{exp}\left( {i\frac{{\phi_1^{(j )} + ({\phi_2^{(j )} + \varepsilon } )}}{2}} \right)\cos \frac{{\phi _1^{(j )} - ({\phi_2^{(j )} + \varepsilon } )}}{2}}\\ { = \textrm{exp}\left( {i\frac{{\phi_1^{(j )} + \phi_2^{(j )}}}{2}} \right)\left[ {({1 + i\varepsilon } )\cos \frac{{\phi_1^{(j )} - \phi_2^{(j )}}}{2} + \varepsilon \sin \frac{{\phi_1^{(j )} - \phi_2^{(j )}}}{2}} \right]} \end{array}$$
up to the first order of ɛ. The phase shift error results in the change of the amplitude ratio and the relative phase. In a widely used direct phase modulator, the phase shift error causes the relative phase shift linear to ɛ, as the phase varies linearly on the applied voltage.

To reduce the effect of voltage shifts, we propose using a dual parallel modulator (DPM) [27] for state preparation. The DPM is widely used as IQ modulator in coherent optical communications. As shown in Fig. 3, the DPM is constructed with two push–pull Mach–Zehnder modulators connected in parallel. In each of the two modulators, an electrode provides phase shifts to the two branches, where the shifts are equal in magnitude but with opposite sign. The phase shifter shown as block i in Fig. 3 provides a fixed π/2 phase shift to the output of the second modulator. The electric field of the output light, ${E_\textrm{o}}$, is related to that of the input light, ${E_\textrm{i}}$, as follows:

$${E_o} = \frac{{\cos {\phi _1} + i\cos {\phi _2}}}{{\sqrt 2 }}{E_i}.$$
Then the photon state is given by:
$$|{\Psi}\rangle = \frac{1}{{\sqrt {\mathop \sum \nolimits_{j = 0.1} ({{{\cos }^2}{\phi_1}^{(j )} + {{\cos }^2}{\phi_2}^{(j )}} )} }}\mathop \sum \nolimits_{j = 0,1} ({\cos {\phi_1}^{(j )} + i\cos {\phi_2}^{(j )}} )|j\rangle.$$
Then, the four states, X0, X1, Y0, and Y1, ${|\psi}\rangle = [{|{0\rangle + \textrm{exp}({i\theta } )} |1\rangle } ]/\sqrt 2, \; \theta \in \{{0, {\pi}/2, {\pi}, \;3{\pi}/2} \}$, can be obtained by adjusting voltages ${{V}_{{\phi _{1}}}}$ and ${{V}_{{\phi _{2}}}}$ to provide the appropriate phase shifts, ${\phi _\textrm{1}}$ and ${\phi _\textrm{2}}$, respectively, as shown in Table 2. Here global phase is ignored. The DPM is operated with two phase shifts of 0 and π on each interferometer to generate the four states, and thus only two output values of 0 and ${{V}_{\pi}}$ are required, simplifying the driver.

 figure: Fig. 3.

Fig. 3. Schematic of DPM (${E_\textrm{i}}$, electric field of input light; ${V_\phi }$, applied voltage to electrodes; $\phi $, phase shift; ${E_\textrm{o}}$, electric field of output light; MZM, Mach–Zehnder modulator; i, $\pi /2$ phase shift). DPM is constructed with two push–pull Mach–Zehnder modulators connected in parallel.

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Tables Icon

Table 2. Phase shifts in DPM to generate photon states X0, X1, Y0, and Y1. Variable $\phi _{i}^{({j} )}$ (i = 1, 2, j = 0, 1) denotes phase applied to i-th electrode for state $|{j}\rangle$.

Let us consider the relative phase error in the DPM, where voltage shift ${{V}_{\boldsymbol \varepsilon }}$ on ${{V}_{{\phi _\textrm{2}}}}$ produces phase shift error ɛ. As the cosine varies proportionally to the square of the error for ${\phi _\textrm{2}}$ = 0 or π:

$$\cos ({\phi _2} + \varepsilon ) = \cos {\phi _2}(1 - \frac{1}{{2!}}{\varepsilon ^2} + \frac{1}{{4!}}{\varepsilon ^4} - \cdots ),$$
Therefore, the lowest-order error during phase modulation is proportional to the square of phase shift error ɛ. When error ɛ is small enough (i.e., much lower than 1), the relative phase error by the DPM is negligible. Hence, the state preparation flaw is suppressed in the DPM much more effectively than in the DDM.

3. Experimental state characterization

We experimentally investigated the effect of voltage shifts on quantum states.

Figure 4 illustrates the experimental setup to investigate the effects of voltage shifts on the photon states of a phase-encoding BB84 QKD system. At Alice’s station, a laser diode emitted light pulses of 1550 nm wavelength, 625MHz and 100 ps duration. The pulse was converted into a pair of coherent double pulses with 800 ps separation using a PLC-AMZI with 800 ps time delay. The double pulse entered either the DDM or DPM to modulate the relative phase. The amount of modulation was determined by electrical pulses (AC) delivered by pulse pattern generators and DC biases (DC). We changed the applied voltage at one electrode from the designed values from –0.25${{V}_{\pi}}$ to 0.25${{V}_{\pi}}$ (i.e., ɛ ${\in} $ {–0.25π, 0.25π}), while the voltage applied to the other electrode was fixed to the designed value. We separately examined the effects of bias drift (denoted as DC) and pulse height deviation (denoted as AC), as shown in Fig. 4. The difference between two types is that both pulse $|0\rangle$ and $|1\rangle$ are influenced by the change of DC bias, it means that the phase shift $\varepsilon $ occurs on $\phi _2^{(0 )}$ and $\phi _2^{(1 )}$, while there is only an impact on the pulse $|1\rangle$ ($\phi _2^{(1 )}$) for the change of the pulse height from PPG. We also fixed the voltage shifts during measurements to maintain a constant phase error at each measurement point.

 figure: Fig. 4.

Fig. 4. Experimental setup to investigate the effect of voltage shifts on phase-encoding BB84 QKD system (Iso, isolator; PC, polarization controller; PLC, planar light-wave circuit; PM, phase modulator; PPG, pulse pattern generator; ${V_\phi }$, applied voltage to electrodes).

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To measure the states of light, another planar light-wave circuit AMZI was used at Bob’s station. We selected the measurement basis X or Y by changing the relative phase between the arms in the AMZI according to temperature [21]. We adjusted the time delay in the interferometers by changing temperature to obtain the highest visibilities when the applied voltages were set to the designed values. Thus, we assumed that the measurement bases were aligned perfectly. We used a MEMS optical switch, with <−60dB crosstalk, to choose the detection port from the two output ports of the AMZI, i.e., bit value 0 or 1, and measured the light intensity with a high-speed photodetector, with 10 GHz bandwidth, instead of using two photodetectors connecting to the two ports. This prevented errors caused by the different characteristics between photodetectors. We measured 18 pulses in 0(1)-port and discard the measurement results during the transition time of the optical switch (0.5ms). Among the three time components in the output of Bob’s AMZI, we measured only the center one, which brings information of the relative phase between $\textrm{|0}\rangle $ and $\textrm{|1}\rangle $. We recorded the intensities of the interference signals with an oscilloscope of 6 GHz bandwidth and 20 GSa/s sampling rate. The intensity was defined as the pulse area. Since we used only the ratio of the intensities, the absolute accuracy of the intensities was not required. We also measured the states in the Z basis to perform quantum state tomography [28]. The measurements in the Z basis refer to the intensity of each component in the double pulses after the phase modulator in Alice’s station. The measurements were done with unattenuated light. Therefore, the measured intensities were proportional to the expectation values, considering the attenuation as independent from the photon states. This assumption is natural, because the characteristics of passive attenuators are expected to remain unchanged during the period of double-pulse components and keep the relative phase.

The state preparation flaw is described by the fidelity between the X- and Y-basis states calculated using the corresponding density matrices as shown in Eq. (1). The density matrices can be reconstructed by state tomography and experimental results. In the following, we describe the estimation of the density matrix of Alice’s states in the X basis, ${{\rho}_\textrm{X}}$. We can estimate the density matrix in the Y basis, ${{\rho}_\textrm{Y}}$, similarly. When Alice emits ${|\textrm{X}i^{\prime}\rangle}$ (i = 0, 1), a state that she believes to be ideal, it is given by

$${\rho _\textrm{X}} = \frac{1}{2}(|{{\mathop{\rm X}\nolimits} 0^{\prime}} \rangle \left\langle {{\mathop{\rm X}\nolimits} 0^{\prime}} \right|+ |{{\mathop{\rm X}\nolimits} 1^{\prime}} \rangle \left\langle {{\mathop{\rm X}\nolimits} 1^{\prime}} \right|).$$
In the following, we assume that single-photon states sent by Alice are described by a matrix in the two-dimensional Hilbert space spanned by the computational basis $\{{{|0\rangle, |1\rangle}} \} = \{{\textrm{|Z0}\rangle, \textrm{|Z1}\rangle} \}$, because we are interested in the distinguishability for such states. Multi-photon states are assumed to be distinguishable as in the conventional decoy method.

This assumption is plausible provided that we can identically control the light fields of the four states in terms of degrees of freedom other than the relative phase, such as amplitude, frequency, timing, polarization, and spatial mode. Then, the density matrix can be expanded by Pauli matrices denoted by ${\sigma _i}$ as

$$\rho = \frac{1}{2}\sum\limits_{i = 0}^3 {\frac{{{S_i}}}{{{S_0}}}{\sigma _i}} ,$$
where ${S_i}$ represents Stokes parameters given by
$${S_0} = 2{n_0},$$
$${S_1} = 2({n_1} - {n_0}),$$
$${S_2} = 2({n_2} - {n_0}),$$
$${S_3} = 2({n_3} - {n_0}).$$
The numbers ${n_i}(i = 0,1,2,3)$ are related to the density matrix and measurement results as follows:
$${n_0} = \frac{N}{2}(\left\langle 0 \right|{\rho _X}|0 \rangle + \left\langle 1 \right|{\rho _X}|1 \rangle ) = \frac{N}{2}(P({{\mathop{\rm Z}\nolimits} 0|{\mathop{\rm X}\nolimits} 0^{\prime}} )+ P({{\mathop{\rm Z}\nolimits} 0|{\mathop{\rm X}\nolimits} 1^{\prime}} )+ P({{\mathop{\rm Z}\nolimits} 1|{\mathop{\rm X}\nolimits} 0^{\prime}} )+ P({{\mathop{\rm Z}\nolimits} 1|{\mathop{\rm X}\nolimits} 1^{\prime}} )),$$
$${n_1} = \frac{N}{2}(\left\langle 0 \right|{\rho _X}|0 \rangle ) = N(P({{\mathop{\rm Z}\nolimits} 0|{\mathop{\rm X}\nolimits} 0^{\prime}} )+ P({{\mathop{\rm Z}\nolimits} 0|{\mathop{\rm X}\nolimits} 1^{\prime}} )),$$
$${n_2} = \frac{N}{2}(\left\langle {{\mathop{\rm X}\nolimits} 1} \right|{\rho _X}|{{\mathop{\rm X}\nolimits} 1} \rangle ) = N(P({{\mathop{\rm X}\nolimits} 1|{\mathop{\rm X}\nolimits} 0^{\prime}} )+ P({{\mathop{\rm X}\nolimits} 1|{\mathop{\rm X}\nolimits} 1^{\prime}} )),$$
$${n_3} = \frac{N}{2}(\left\langle {{\mathop{\rm Y}\nolimits} 1} \right|{\rho _X}|{{\mathop{\rm Y}\nolimits} 1} \rangle = N(P({{\mathop{\rm Y}\nolimits} 1|{\mathop{\rm X}\nolimits} 0^{\prime}} )+ P({{\mathop{\rm Y}\nolimits} 1|{\mathop{\rm X}\nolimits} 1^{\prime}} )),$$
where N is a constant dependent on the detector efficiency and light intensity, P$(\textrm{Z0|X0}^{\prime}) = \langle\textrm{Z0|X0}^{\prime}\rangle\langle\textrm{X0}^{\prime}|\textrm{Z0}\rangle$ is the probability that the measurement result is 0 in the Z basis when state $\textrm{|X0}^{\prime}\rangle$ is generated by Alice, and the other probabilities are defined similarly. The probabilities required to obtain ${{n}_{i}}$ were calculated from the measured intensities. For example:
$$P({\mathop{\rm Z}\nolimits} 0|{\mathop{\rm X}\nolimits} 0^{\prime}) = \frac{{I({\mathop{\rm Z}\nolimits} 0|{\rm X}0^{\prime})}}{{I({\mathop{\rm Z}\nolimits} 0|{\rm X}0^{\prime}) + I({\mathop{\rm Z}\nolimits} 1|{\rm X}0^{\prime})}},$$
where I$({\textrm{Z0|X0}^{\prime}} )$ and I$({\textrm{Z1|X0}^{\prime}} )$ represent the intensities of pulses measured in photodetectors Z0 and Z1, respectively, when Alice generates state $\textrm{|X0}^{\prime}\rangle$ while intending to send state $\textrm{|X0}\rangle$.

4. Results and discussion

Figure 5 shows the reconstructed density matrices ${{\rho}_\textrm{X}}$ for X states. They are reconstructed by the intensities of the interference signals and substituted into Eq. (12)–(21). Experimental details are shown in the section 3. Figures 5(a) and 5(d) show that, without voltage shift, the density matrices from the DDM and DPM are both close to half of the identity matrix, which implies that the states are almost ideal [28]. However, the states generated by the DDM in Figs. 5(b) and 5(c), deviate from the ideal one by changing 25% of ${{V}_{\pi}}$ in the applied AC or DC voltage, whereas those generated by the DPM remain close to ideal as shown in Figs. 5(e) and 5(f).

 figure: Fig. 5.

Fig. 5. Reconstructed state ${\rho _X}$ with DDM (a) without voltage variation and with (b) 25% DC bias and (c) 25% AC amplitude change. Reconstructed state ${\rho _X}$ with DPM (d) without voltage variation and with (e) 25% DC bias and (f) 25% AC amplitude change.

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The effects of the modulation voltage on fidelity F(${{\rho}_\textrm{X}},{{\rho}_\textrm{Y}}$) are summarized in Figs. 6(a) and 6(b) for the DC bias and AC amplitude changes, respectively. We substituted the reconstructed density matrices into Eq. (2) to obtain the fidelity. The error bars mainly originate from the noise of the oscilloscope during the measurement of output states. The observed signal-to-noise ratio was approximately 16 dB. The effect of this estimation error was negligible on the final key generation rate, as verified in the simulation reported below. The fidelity of sent states using the DPM remained almost at unity for the voltage deviation range in the present experiment, whereas the states generated by the DDM were affected by the voltage deviation. These results agree well with the theoretical prediction. A small decrease observed under a large voltage deviation may result from the nonlinear response of the phase modulators.

 figure: Fig. 6.

Fig. 6. (a) Fidelity according to DC bias. (b) Fidelity according to pulse amplitude (AC). Squares and stars represent the measured values for DDM and DPM, respectively. The solid and dotted lines represent the theoretical values for DDM and DPM, respectively.

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We then evaluated the QKD system performance by simulating the final key generation rate considering the state preparation flaw for decoy state BB84 protocol in the asymptotic case. We use experimentally obtained fidelity and visibility to express the state preparation flaw as described in section 2. For the simulation, we used experimental results described in Sec. 3, Eq. (4) and the parameters listed in Table 3, and the error rates were calculated for signal (s) and decoy (d) pulses in the X and Y bases, as follows [2325]:

$${e_{i, j}} = \frac{1}{{{S_{{\mu _j}}}}}\left[ {\left( {1 - \exp \left( { - \eta {\mu_j}{{10}^{ - \frac{{\alpha l}}{{10}}}}} \right)} \right){E_i} + \frac{1}{2}\exp \left( { - \eta {\mu_j}{{10}^{ - \frac{{\alpha l}}{{10}}}}} \right){y_0}} \right], i = \textrm{X},\textrm{Y}, j = \textrm{d},\textrm{s}$$
where η is the detector efficiency, α (in dB/km) is the loss coefficient in fiber, l is the transmission distance, ${y_0}$ is the background rate, ${\mu _\textrm{s}}$ and ${\mu _\textrm{d}}$ are the mean photon numbers of the signal and nontrivial decoy state, respectively, ${{E}_\textrm{i}}$ is the source error rate for the corresponding basis, ${{S}_{{{\mu}_\textrm{d}}}}$ is the count rate of the decoy pulse, and ${{S}_{{{\mu}_\textrm{s}}}}$ is the count rate of signal pulse:
$${S_{{\mu _\textrm{d}}}} = 1 + ( - 1 + {y_0})\exp \left( { - \eta {\mu_\textrm{d}}{{10}^{ - \frac{{\alpha l}}{{10}}}}} \right),$$
$${S_{{\mu _\textrm{s}}}} = 1 + ( - 1 + {y_0})\exp \left( { - \eta {\mu_\textrm{s}}{{10}^{ - \frac{{\alpha l}}{{10}}}}} \right).$$
Source error rate ${{E}_{i}}$ (i = X, Y) represents the rate at which Alice generates states with incorrect bit values in the corresponding basis and is related to visibility ${\nu _{i}}$ of the interference by
$${E_i} = \frac{{1 - {v_i}}}{2}.$$
The visibilities can be calculated from the measured intensities as
$${v_{\mathop{\rm X}\nolimits} } = \frac{{I({\mathop{\rm X}\nolimits} 0|{\rm X}0^{\prime}) - I({\mathop{\rm X}\nolimits} 1|{\rm X}0^{\prime})}}{{I({\mathop{\rm X}\nolimits} 0|{\rm X}0^{\prime}) + I({\mathop{\rm X}\nolimits} 1|{\rm X}0^{\prime})}},$$
$${v_{\mathop{\rm Y}\nolimits} } = \frac{{I({\mathop{\rm Y}\nolimits} 0|{\rm Y}0^{\prime}) - I({\mathop{\rm Y}\nolimits} 1|{\rm Y}0^{\prime})}}{{I({\mathop{\rm Y}\nolimits} 0|{\rm Y}0^{\prime}) + I({\mathop{\rm Y}\nolimits} 1|{\rm Y}0^{\prime})}}.$$
where I$({\textrm{X0|X0}^{\prime}} )$ and I$({\textrm{X1|X0}^{\prime}} )$ represent the intensities of pulses measured in photodetectors X0 and X1, respectively, when Alice generates state $\textrm{|X0}^{\prime}\rangle$ while intending to send state $\textrm{|X0}\rangle$. In our decoy state method to calculate the secure key generated from signal pulses, we used quantum bit error rate (QBER) ${\delta _\textrm{X}} = {e_{\textrm{X}, s}}$ calculated with Eq. (22). Using vacuum + weak decoy states [23], we consider the single-photon state error rate,
$${e_1} \le \frac{{{e_{{\rm Yd}}}{e^{{\mu _{\rm d}}}} - {e_0}{y_0}}}{{y_1^{L,{\mu _{\rm d}},0}{\mu _{\rm d}}}},$$
as phase error rate ${\delta _\textrm{Y}}$. To account for the influence of the state preparation flaw $\Delta $, the fidelity from the experimentally obtained density matrices with Eq. (2) is used in Eq. (1). We calculate ${\delta^{\prime}_\textrm{Y}}$ as Eq. (3) and substitute it into Eq. (4) to obtain the final key. Here, ${\textrm{e}_\textrm{0}}$ is the error rate of zero-photon state, and we used the same value of ${{e}_{0}}{{y}_{0}}$ as in [23]. The yield of the single-photon state is given by
$$y_1^{L,{\mu _{\rm d}},0} = \frac{{{\mu _{\rm s}}}}{{{\mu _{\rm d}}{\mu _{\rm s}} - {\mu _{\rm d}}^2}}({Q_\textrm{d}}{e^{{\mu _{\rm s}}}} - {Q_\textrm{s}}{e^{{\mu _{\rm d}}}}\frac{{{\mu _{\rm d}}^2}}{{{\mu _{\rm s}}^2}} - \frac{{{\mu _{\rm s}}^2 - {\mu _{\rm d}}^2}}{{{\mu _{\rm s}}^2}}{y_0}),$$
where the superscript in $\textrm{$\textit{y}$}_\textrm{1}^{\textrm{$\textit{L}$},{{\mu}_\textrm{d}},\textrm{0}}$ represents distance, mean photon number of the weak decoy state, and mean photon number of the vacuum decoy state. The gains can be calculated as above with the given parameters as
$${Q_1} \ge Q_1^{L,{\mu _{\rm d}},0} = \frac{{\mu _{\rm s}^2}}{{{\mu _{\rm d}}{\mu _{\rm s}} - {\mu _{\rm d}}^2}}({S_{{\mu _{\rm d}}}}{e^{{\mu _{\rm s}}}} - {S_{{\mu _{\rm s}}}}{e^{{\mu _{\rm d}}}}\frac{{{\mu _{\rm d}}^2}}{{{\mu _{\rm s}}^2}} - \frac{{{\mu _{\rm s}}^2 - {\mu _{\rm d}}^2}}{{{\mu _{\rm s}}^2}}{y_0}),$$

Tables Icon

Table 3. Parameters for key generation rate

The simulation results are shown in Fig. 7 and clearly describe the effects of the state preparation flaw. In both the DC bias and AC amplitude changes, the DPM yields higher key generation rate than the DDM. The final key generation rate is almost unchanged up to 0.2${{V}_{\pi}}$ deviation for AC, and a finite key generation rate is obtained at 180 km under DC bias variation. As the DC bias changes the phases of both states $\textrm{|0}\rangle$ and $\textrm{|1}\rangle$, its effect is more notable, though the errors in DC bias would be much smaller than AC deviation in a practical device. As our experimental fidelity and simulation result, state preparation flaw generated by the imperfection in implementation will lead to a decrease in key generation rate as theoretical analysis [2], [1415]. The improvement on intensity modulator and phase modulator is also studied to decrease the imperfection in implementation [29]. Our proposal take advantage of the PLC-AMZI to keep the stable interference to avoid inaccuracy during the state preparation. Moreover, our proposal, DPM, is an off-the-shelf device used in optical communication. It requires only two values, 0 and Vπ, to define the BB84 states will simplify the design of the driving circuit. Other researches based on the feedback of the quantum key distribution system were also supplied to decrease the imperfection in implementation [30]. Our proposal can suppress the state preparation flaw without complicated feedback circuits. Of course, the combination of these researches and ours will decrease state preparation flaw further and provide a much more efficient transmitter.

 figure: Fig. 7.

Fig. 7. Final key generation rate by (a) DC and (b) AC variations in the modulator. Dashed lines represent voltage deviation of 0 (red), 0.05${{V}_{\pi}}$ (blue) and 0.1${{V}_{\pi}}$ (black) in DDM. Solid lines represent voltage deviation of 0 (red), 0.05${{V}_{\pi}}$ (blue), and 0.1${{V}_{\pi}}$ (black) and 0.2${{V}_{\pi}}$ (green)in DPM.

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5. Conclusion

We investigated the effects of the state preparation flaw appearing from relative phase errors caused by the limited bandwidth of the electric driving circuits in high-speed BB84 QKD systems. Although signal waveform degradation is common in high-speed communication systems, the relative phase error in QKD systems may increase the eavesdropper access to information, severely reducing the final key generation rate. To mitigate this risk, we propose phase modulation using a DPM to prepare BB84 states immune to signal degradation. We quantitively evaluated the state preparation flaw in terms of fidelity between the states of conjugated bases. We estimated the density matrices of the generated light states using state tomography and demonstrated that the fidelity between the conjugated states remain close to unity under voltage shifts up to 25%. A simulation of the final key generation rate according to transmission distance shows a negligible decrease for voltage shifts up to 20%. This high tolerance and the fact that the DPM requires only two values, 0 and Vπ, to define the BB84 states simplify the design of the driving circuit. Moreover, the necessity of fine tuning the driving voltage during manufacturing decreases. Therefore, the proposed use of the DPM for state preparation can support implementation security and reduce production costs. We believe that our proposal is conductive to the widespread deployment of QKD systems.

Funding

Council for Science, Technology and Innovation; Japan Society for the Promotion of Science; Hokkaido University.

Acknowledgments

The authors would like to thank Mr. K. Nakata for his support during this work. This research was supported by CSTI SIP "Photonics and Quantum Technology for Society 5.0" (Funding agency: QST), JSPS KAKENHI under Grant Number 18H05237, and Global Station for Big Data and Cybersecurity, a project of Global Institution for Collaborative Research and Education at Hokkaido University.

Disclosures

The authors declare no conflicts of interest.

References

1. C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proc. IEEE Int. Conf. Comput. Syst. Signal Process., Bangalore, India, 175–179 (1984).

2. H.-K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014). [CrossRef]  

3. A. R. Dixon, Z. L. Yuan, J. F. Dynes, A. W. Sharpe, and A. J. Shields, “Gigaherts decoy quantum key distribution with 1 Mbit/s secure key rate,” Opt. Express 16(23), 18790–18797 (2008). [CrossRef]  

4. S. Wang, W. Chen, J. Guo, Z. Yin, H. Li, Z. Zhou, G. Guo, and Z. Han, “2 GHz clock quantum key distribution over 260 km of standard telecom fiber,” Opt. Lett. 37(6), 1008–1010 (2012). [CrossRef]  

5. S. Wang, W. Chen, Z. Yin, D. He, C. Hui, P. Hao, G. Yuan, C. Wang, L. Zhang, J. Kuang, S. Liu, Z. Zhou, Y. Wang, G. Guo, and Z. Han, “Practical gigahertz quantum key distribution robust against channel disturbance,” Opt. Lett. 43(9), 2030–2033 (2018). [CrossRef]  

6. S. Wang, D. He, Z. Yin, F. Lu, C. Cui, W. Chen, Z. Zhou, G. Guo, and Z. Han, “Beating the fundamental rate-distance limit in a proof-of-principle quantum key distribution system,” Phys. Rev. X 9(2), 021046 (2019). [CrossRef]  

7. R. J. Hughes, G. L. Morgan, and C. G. Peterson, “Quantum key distribution over a 48 km optical fibre network,” J. Mod. Opt. 47(2–3), 533–547 (2000). [CrossRef]  

8. M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009). [CrossRef]  

9. M. Sasaki, M. Fujiwara, H. Ishizuka, W. Klaus, K. Wakui, M. Takeoka, S. Miki, T. Yamashita, Z. Wang, A. Tanaka, K. Yoshino, Y. Nambu, S. Takahashi, A. Tajima, A. Tomita, T. Domeki, T. Hasegawa, Y. Sakai, H. Kobayashi, T. Asai, K. Shimizu, T. Tokura, T. Tsurumaru, M. Matsui, T. Honjo, K. Tamaki, H. Takesue, Y. Tokura, J. F. Dynes, A. R. Dixon, A. W. Sharpe, Z. L. Yuan, A. J. Shields, S. Uchikoga, M. Legré, S. Robyr, P. Trinkler, L. Monat, J.-B. Page, G. Ribordy, A. Poppe, A. Allacher, O. Maurhart, T. Länger, M. Peev, and A. Zeilinger, “Field test of quantum key distribution in the Tokyo QKD network,” Opt. Express 19(11), 10387–10409 (2011). [CrossRef]  

10. D. Stucki, M. Legré, F. Buntschu, B. Clausen, N. Felber, N. Gisin, L. Henzen, P. Junod, G. Litzistorf, P. Monbaron, L. Monat, J.-B. Page, D. Perroud, G. Ribordy, A. Rochas, S. Robyr, J. Tavares, R. Thew, P. Trinkler, S. Ventura, R. Voirol, N. Walenta, and H. Zbinden, “Long-term performance of the SwissQuantum quantum key distribution network in a field environment,” New J. Phys. 13(12), 123001 (2011). [CrossRef]  

11. S. Wang, W. Chen, Z. Yin, H. Li, D. He, Y. Li, Z. Zhou, X. Song, F. Li, D. Wang, H. Chen, Y. Han, J. Huang, J. Guo, P. Hao, M. Li, C. Zhang, D. Liu, W. Liang, C. Miao, P. Wu, G. Guo, and Z. Han, “Field and long-term demonstration of a wide area quantum key distribution network,” Opt. Express 22(18), 21739–21756 (2014). [CrossRef]  

12. Y. Mao, B.-X. Wang, C. Zhao, G. Wang, R. Wang, H. Wang, F. Zhou, J. Nie, Q. Chen, Y. Zhao, Q. Zhang, J. Zhang, T.-Y. Chen, and J.-W. Pan, “Integrating quantum key distribution with classical communications in backbone fiber network,” Opt. Express 26(5), 6010–6020 (2018). [CrossRef]  

13. P. W. Shor and J. Preskill, “Simple proof of the BB84 quantum key distribution system,” Phys. Rev. Lett. 85(2), 441–444 (2000). [CrossRef]  

14. K. Tamaki, M. Curty, and M. Lucamarani, “Decoy-state quantum key distribution with a leaky source,” New J. Phys. 18(6), 065008 (2016). [CrossRef]  

15. D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, “Security of quantum key distribution with imperfect devices,” Quantum Inf. Comput. 4(5), 135 (2004). [CrossRef]  

16. K. Yoshino, M. Fujiwara, K. Nakata, T. Sumiya, T. Sasaki, M. Takeoka, M. Sasaki, A. Tajima, M. Koashi, and A. Tomita, “Quantum key distribution with an efficient countermeasure against correlated intensity fluctuations in optical pulses,” Npj Quantum Inf. 4(1), 8 (2018). [CrossRef]  

17. S. M. Barnett, B. Huttner, and S. J. Phoenix, “Eavesdropping strategies and rejected-data protocols in quantum cryptography,” J. Mod. Opt. 40(12), 2501–2513 (1993). [CrossRef]  

18. K. Tamaki, M. Curty, G. Kato, H.-K. Lo, and K. Azuma, “Loss-tolerant quantum cryptography with imperfect sources,” Phys. Rev. A 90(5), 052314 (2014). [CrossRef]  

19. G. Bonfrate, M. Harlow, C. Ford, G. Maxwell, and P. D. Townsend, “Asymmetric Mach-Zehnder germane-silicate channel waveguide interferometers for quantum cryptography systems,” Electron. Lett. 37(13), 846–847 (2001). [CrossRef]  

20. T. Kimura, Y. Nambu, T. Hatanaka, A. Tomita, H. Kosaka, and K. Nakamura, “Single-photon interference over 150 km Transmission using silica-based integrated-optic interferometers for quantum cryptography,” Jpn. J. Appl. Phys. 43(No. 9A/B), L1217–L1219 (2004). [CrossRef]  

21. Y. Nambu, T. Hatanaka, and K. Nakamura, “B84 quantum key distribution system based on silica-based planar lightwave circuits,” Jpn. J. Appl. Phys. 43(No. 8B), L1109–L1110 (2004). [CrossRef]  

22. K. Yoshino, A. Tanaka, Y. Nambu, A. Tajima, and A. Tomita, “Dual-mode time-bin coding for quantum key distribution using dual-drive Mach-Zehnder modulator,” IEEE Eur. Conf. Exhibit. Opt. Commun., Berlin, Germany, 1–2 (2007).

23. X. Ma, B. Qi, Y. Zhao, and H.-K. Lo, “Practical decoy state for quantum key distribution,” Phys. Rev. A 72(1), 012326 (2005). [CrossRef]  

24. W.-Y. Hwang, “Quantum key distribution with high loss: toward global secure communication,” Phys. Rev. Lett. 91(5), 057901 (2003). [CrossRef]  

25. X.-B. Wang, “Beating the photon-number-splitting attack in practical quantum cryptography,” Phys. Rev. Lett. 94(23), 230503 (2005). [CrossRef]  

26. H.-K. Lo, H. F. Chau, and M. Ardehali, “Efficient quantum key distribution scheme and proof of its unconditional security,” J. Cryptology 18(2), 133–165 (2005). [CrossRef]  

27. T. Kawanishi, T. Sakamoto, A. Chiba, M. Izutsu, K. Higuma, J. Ichikawa, T. Lee, and V. Filsinger, “High-speed dual-parallel Mach-Zehnder modulator using thin lithium niobate substrate,” Proc. OFC/NFOEC, San Diego, CA, USA, 1–3 (2008).

28. D. F. V. James, P. Kwiat, W. Munro, and A. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312 (2001). [CrossRef]  

29. G. L. Roberts, M. Pittaluga, M. Minder, M. Lucamarini, J. F. Dynes, Z. L. Yuan, and A. J. Shields, “Pattern-effect-free intensity modulator for secure decoy-state quantum key distribution,” Opt. Lett. 43(20), 5110–5113 (2018). [CrossRef]  

30. J. Liu, H. Ding, C. Zhang, S. Xie, and Q. Wang, “Practical Phase-Modulation Stabilization in Quantum Key Distribution via Machine learning,” Phys. Rev. Appl. 12(1), 014059 (2019). [CrossRef]  

References

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  1. C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proc. IEEE Int. Conf. Comput. Syst. Signal Process., Bangalore, India, 175–179 (1984).
  2. H.-K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014).
    [Crossref]
  3. A. R. Dixon, Z. L. Yuan, J. F. Dynes, A. W. Sharpe, and A. J. Shields, “Gigaherts decoy quantum key distribution with 1 Mbit/s secure key rate,” Opt. Express 16(23), 18790–18797 (2008).
    [Crossref]
  4. S. Wang, W. Chen, J. Guo, Z. Yin, H. Li, Z. Zhou, G. Guo, and Z. Han, “2 GHz clock quantum key distribution over 260 km of standard telecom fiber,” Opt. Lett. 37(6), 1008–1010 (2012).
    [Crossref]
  5. S. Wang, W. Chen, Z. Yin, D. He, C. Hui, P. Hao, G. Yuan, C. Wang, L. Zhang, J. Kuang, S. Liu, Z. Zhou, Y. Wang, G. Guo, and Z. Han, “Practical gigahertz quantum key distribution robust against channel disturbance,” Opt. Lett. 43(9), 2030–2033 (2018).
    [Crossref]
  6. S. Wang, D. He, Z. Yin, F. Lu, C. Cui, W. Chen, Z. Zhou, G. Guo, and Z. Han, “Beating the fundamental rate-distance limit in a proof-of-principle quantum key distribution system,” Phys. Rev. X 9(2), 021046 (2019).
    [Crossref]
  7. R. J. Hughes, G. L. Morgan, and C. G. Peterson, “Quantum key distribution over a 48 km optical fibre network,” J. Mod. Opt. 47(2–3), 533–547 (2000).
    [Crossref]
  8. M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
    [Crossref]
  9. M. Sasaki, M. Fujiwara, H. Ishizuka, W. Klaus, K. Wakui, M. Takeoka, S. Miki, T. Yamashita, Z. Wang, A. Tanaka, K. Yoshino, Y. Nambu, S. Takahashi, A. Tajima, A. Tomita, T. Domeki, T. Hasegawa, Y. Sakai, H. Kobayashi, T. Asai, K. Shimizu, T. Tokura, T. Tsurumaru, M. Matsui, T. Honjo, K. Tamaki, H. Takesue, Y. Tokura, J. F. Dynes, A. R. Dixon, A. W. Sharpe, Z. L. Yuan, A. J. Shields, S. Uchikoga, M. Legré, S. Robyr, P. Trinkler, L. Monat, J.-B. Page, G. Ribordy, A. Poppe, A. Allacher, O. Maurhart, T. Länger, M. Peev, and A. Zeilinger, “Field test of quantum key distribution in the Tokyo QKD network,” Opt. Express 19(11), 10387–10409 (2011).
    [Crossref]
  10. D. Stucki, M. Legré, F. Buntschu, B. Clausen, N. Felber, N. Gisin, L. Henzen, P. Junod, G. Litzistorf, P. Monbaron, L. Monat, J.-B. Page, D. Perroud, G. Ribordy, A. Rochas, S. Robyr, J. Tavares, R. Thew, P. Trinkler, S. Ventura, R. Voirol, N. Walenta, and H. Zbinden, “Long-term performance of the SwissQuantum quantum key distribution network in a field environment,” New J. Phys. 13(12), 123001 (2011).
    [Crossref]
  11. S. Wang, W. Chen, Z. Yin, H. Li, D. He, Y. Li, Z. Zhou, X. Song, F. Li, D. Wang, H. Chen, Y. Han, J. Huang, J. Guo, P. Hao, M. Li, C. Zhang, D. Liu, W. Liang, C. Miao, P. Wu, G. Guo, and Z. Han, “Field and long-term demonstration of a wide area quantum key distribution network,” Opt. Express 22(18), 21739–21756 (2014).
    [Crossref]
  12. Y. Mao, B.-X. Wang, C. Zhao, G. Wang, R. Wang, H. Wang, F. Zhou, J. Nie, Q. Chen, Y. Zhao, Q. Zhang, J. Zhang, T.-Y. Chen, and J.-W. Pan, “Integrating quantum key distribution with classical communications in backbone fiber network,” Opt. Express 26(5), 6010–6020 (2018).
    [Crossref]
  13. P. W. Shor and J. Preskill, “Simple proof of the BB84 quantum key distribution system,” Phys. Rev. Lett. 85(2), 441–444 (2000).
    [Crossref]
  14. K. Tamaki, M. Curty, and M. Lucamarani, “Decoy-state quantum key distribution with a leaky source,” New J. Phys. 18(6), 065008 (2016).
    [Crossref]
  15. D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, “Security of quantum key distribution with imperfect devices,” Quantum Inf. Comput. 4(5), 135 (2004).
    [Crossref]
  16. K. Yoshino, M. Fujiwara, K. Nakata, T. Sumiya, T. Sasaki, M. Takeoka, M. Sasaki, A. Tajima, M. Koashi, and A. Tomita, “Quantum key distribution with an efficient countermeasure against correlated intensity fluctuations in optical pulses,” Npj Quantum Inf. 4(1), 8 (2018).
    [Crossref]
  17. S. M. Barnett, B. Huttner, and S. J. Phoenix, “Eavesdropping strategies and rejected-data protocols in quantum cryptography,” J. Mod. Opt. 40(12), 2501–2513 (1993).
    [Crossref]
  18. K. Tamaki, M. Curty, G. Kato, H.-K. Lo, and K. Azuma, “Loss-tolerant quantum cryptography with imperfect sources,” Phys. Rev. A 90(5), 052314 (2014).
    [Crossref]
  19. G. Bonfrate, M. Harlow, C. Ford, G. Maxwell, and P. D. Townsend, “Asymmetric Mach-Zehnder germane-silicate channel waveguide interferometers for quantum cryptography systems,” Electron. Lett. 37(13), 846–847 (2001).
    [Crossref]
  20. T. Kimura, Y. Nambu, T. Hatanaka, A. Tomita, H. Kosaka, and K. Nakamura, “Single-photon interference over 150 km Transmission using silica-based integrated-optic interferometers for quantum cryptography,” Jpn. J. Appl. Phys. 43(No. 9A/B), L1217–L1219 (2004).
    [Crossref]
  21. Y. Nambu, T. Hatanaka, and K. Nakamura, “B84 quantum key distribution system based on silica-based planar lightwave circuits,” Jpn. J. Appl. Phys. 43(No. 8B), L1109–L1110 (2004).
    [Crossref]
  22. K. Yoshino, A. Tanaka, Y. Nambu, A. Tajima, and A. Tomita, “Dual-mode time-bin coding for quantum key distribution using dual-drive Mach-Zehnder modulator,” IEEE Eur. Conf. Exhibit. Opt. Commun., Berlin, Germany, 1–2 (2007).
  23. X. Ma, B. Qi, Y. Zhao, and H.-K. Lo, “Practical decoy state for quantum key distribution,” Phys. Rev. A 72(1), 012326 (2005).
    [Crossref]
  24. W.-Y. Hwang, “Quantum key distribution with high loss: toward global secure communication,” Phys. Rev. Lett. 91(5), 057901 (2003).
    [Crossref]
  25. X.-B. Wang, “Beating the photon-number-splitting attack in practical quantum cryptography,” Phys. Rev. Lett. 94(23), 230503 (2005).
    [Crossref]
  26. H.-K. Lo, H. F. Chau, and M. Ardehali, “Efficient quantum key distribution scheme and proof of its unconditional security,” J. Cryptology 18(2), 133–165 (2005).
    [Crossref]
  27. T. Kawanishi, T. Sakamoto, A. Chiba, M. Izutsu, K. Higuma, J. Ichikawa, T. Lee, and V. Filsinger, “High-speed dual-parallel Mach-Zehnder modulator using thin lithium niobate substrate,” Proc. OFC/NFOEC, San Diego, CA, USA, 1–3 (2008).
  28. D. F. V. James, P. Kwiat, W. Munro, and A. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312 (2001).
    [Crossref]
  29. G. L. Roberts, M. Pittaluga, M. Minder, M. Lucamarini, J. F. Dynes, Z. L. Yuan, and A. J. Shields, “Pattern-effect-free intensity modulator for secure decoy-state quantum key distribution,” Opt. Lett. 43(20), 5110–5113 (2018).
    [Crossref]
  30. J. Liu, H. Ding, C. Zhang, S. Xie, and Q. Wang, “Practical Phase-Modulation Stabilization in Quantum Key Distribution via Machine learning,” Phys. Rev. Appl. 12(1), 014059 (2019).
    [Crossref]

2019 (2)

S. Wang, D. He, Z. Yin, F. Lu, C. Cui, W. Chen, Z. Zhou, G. Guo, and Z. Han, “Beating the fundamental rate-distance limit in a proof-of-principle quantum key distribution system,” Phys. Rev. X 9(2), 021046 (2019).
[Crossref]

J. Liu, H. Ding, C. Zhang, S. Xie, and Q. Wang, “Practical Phase-Modulation Stabilization in Quantum Key Distribution via Machine learning,” Phys. Rev. Appl. 12(1), 014059 (2019).
[Crossref]

2018 (4)

2016 (1)

K. Tamaki, M. Curty, and M. Lucamarani, “Decoy-state quantum key distribution with a leaky source,” New J. Phys. 18(6), 065008 (2016).
[Crossref]

2014 (3)

S. Wang, W. Chen, Z. Yin, H. Li, D. He, Y. Li, Z. Zhou, X. Song, F. Li, D. Wang, H. Chen, Y. Han, J. Huang, J. Guo, P. Hao, M. Li, C. Zhang, D. Liu, W. Liang, C. Miao, P. Wu, G. Guo, and Z. Han, “Field and long-term demonstration of a wide area quantum key distribution network,” Opt. Express 22(18), 21739–21756 (2014).
[Crossref]

H.-K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014).
[Crossref]

K. Tamaki, M. Curty, G. Kato, H.-K. Lo, and K. Azuma, “Loss-tolerant quantum cryptography with imperfect sources,” Phys. Rev. A 90(5), 052314 (2014).
[Crossref]

2012 (1)

2011 (2)

2009 (1)

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
[Crossref]

2008 (1)

2005 (3)

X. Ma, B. Qi, Y. Zhao, and H.-K. Lo, “Practical decoy state for quantum key distribution,” Phys. Rev. A 72(1), 012326 (2005).
[Crossref]

X.-B. Wang, “Beating the photon-number-splitting attack in practical quantum cryptography,” Phys. Rev. Lett. 94(23), 230503 (2005).
[Crossref]

H.-K. Lo, H. F. Chau, and M. Ardehali, “Efficient quantum key distribution scheme and proof of its unconditional security,” J. Cryptology 18(2), 133–165 (2005).
[Crossref]

2004 (3)

T. Kimura, Y. Nambu, T. Hatanaka, A. Tomita, H. Kosaka, and K. Nakamura, “Single-photon interference over 150 km Transmission using silica-based integrated-optic interferometers for quantum cryptography,” Jpn. J. Appl. Phys. 43(No. 9A/B), L1217–L1219 (2004).
[Crossref]

Y. Nambu, T. Hatanaka, and K. Nakamura, “B84 quantum key distribution system based on silica-based planar lightwave circuits,” Jpn. J. Appl. Phys. 43(No. 8B), L1109–L1110 (2004).
[Crossref]

D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, “Security of quantum key distribution with imperfect devices,” Quantum Inf. Comput. 4(5), 135 (2004).
[Crossref]

2003 (1)

W.-Y. Hwang, “Quantum key distribution with high loss: toward global secure communication,” Phys. Rev. Lett. 91(5), 057901 (2003).
[Crossref]

2001 (2)

D. F. V. James, P. Kwiat, W. Munro, and A. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312 (2001).
[Crossref]

G. Bonfrate, M. Harlow, C. Ford, G. Maxwell, and P. D. Townsend, “Asymmetric Mach-Zehnder germane-silicate channel waveguide interferometers for quantum cryptography systems,” Electron. Lett. 37(13), 846–847 (2001).
[Crossref]

2000 (2)

P. W. Shor and J. Preskill, “Simple proof of the BB84 quantum key distribution system,” Phys. Rev. Lett. 85(2), 441–444 (2000).
[Crossref]

R. J. Hughes, G. L. Morgan, and C. G. Peterson, “Quantum key distribution over a 48 km optical fibre network,” J. Mod. Opt. 47(2–3), 533–547 (2000).
[Crossref]

1993 (1)

S. M. Barnett, B. Huttner, and S. J. Phoenix, “Eavesdropping strategies and rejected-data protocols in quantum cryptography,” J. Mod. Opt. 40(12), 2501–2513 (1993).
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Allacher, A.

Alléaume, R.

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
[Crossref]

Ardehali, M.

H.-K. Lo, H. F. Chau, and M. Ardehali, “Efficient quantum key distribution scheme and proof of its unconditional security,” J. Cryptology 18(2), 133–165 (2005).
[Crossref]

Asai, T.

Azuma, K.

K. Tamaki, M. Curty, G. Kato, H.-K. Lo, and K. Azuma, “Loss-tolerant quantum cryptography with imperfect sources,” Phys. Rev. A 90(5), 052314 (2014).
[Crossref]

Barnett, S. M.

S. M. Barnett, B. Huttner, and S. J. Phoenix, “Eavesdropping strategies and rejected-data protocols in quantum cryptography,” J. Mod. Opt. 40(12), 2501–2513 (1993).
[Crossref]

Barreiro, C.

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
[Crossref]

Bennett, C. H.

C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proc. IEEE Int. Conf. Comput. Syst. Signal Process., Bangalore, India, 175–179 (1984).

Bonfrate, G.

G. Bonfrate, M. Harlow, C. Ford, G. Maxwell, and P. D. Townsend, “Asymmetric Mach-Zehnder germane-silicate channel waveguide interferometers for quantum cryptography systems,” Electron. Lett. 37(13), 846–847 (2001).
[Crossref]

Bouda, J.

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
[Crossref]

Boxleitner, W.

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
[Crossref]

Brassard, G.

C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proc. IEEE Int. Conf. Comput. Syst. Signal Process., Bangalore, India, 175–179 (1984).

Buntschu, F.

D. Stucki, M. Legré, F. Buntschu, B. Clausen, N. Felber, N. Gisin, L. Henzen, P. Junod, G. Litzistorf, P. Monbaron, L. Monat, J.-B. Page, D. Perroud, G. Ribordy, A. Rochas, S. Robyr, J. Tavares, R. Thew, P. Trinkler, S. Ventura, R. Voirol, N. Walenta, and H. Zbinden, “Long-term performance of the SwissQuantum quantum key distribution network in a field environment,” New J. Phys. 13(12), 123001 (2011).
[Crossref]

Chau, H. F.

H.-K. Lo, H. F. Chau, and M. Ardehali, “Efficient quantum key distribution scheme and proof of its unconditional security,” J. Cryptology 18(2), 133–165 (2005).
[Crossref]

Chen, H.

Chen, Q.

Chen, T.-Y.

Chen, W.

Chiba, A.

T. Kawanishi, T. Sakamoto, A. Chiba, M. Izutsu, K. Higuma, J. Ichikawa, T. Lee, and V. Filsinger, “High-speed dual-parallel Mach-Zehnder modulator using thin lithium niobate substrate,” Proc. OFC/NFOEC, San Diego, CA, USA, 1–3 (2008).

Clausen, B.

D. Stucki, M. Legré, F. Buntschu, B. Clausen, N. Felber, N. Gisin, L. Henzen, P. Junod, G. Litzistorf, P. Monbaron, L. Monat, J.-B. Page, D. Perroud, G. Ribordy, A. Rochas, S. Robyr, J. Tavares, R. Thew, P. Trinkler, S. Ventura, R. Voirol, N. Walenta, and H. Zbinden, “Long-term performance of the SwissQuantum quantum key distribution network in a field environment,” New J. Phys. 13(12), 123001 (2011).
[Crossref]

Cui, C.

S. Wang, D. He, Z. Yin, F. Lu, C. Cui, W. Chen, Z. Zhou, G. Guo, and Z. Han, “Beating the fundamental rate-distance limit in a proof-of-principle quantum key distribution system,” Phys. Rev. X 9(2), 021046 (2019).
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Curty, M.

K. Tamaki, M. Curty, and M. Lucamarani, “Decoy-state quantum key distribution with a leaky source,” New J. Phys. 18(6), 065008 (2016).
[Crossref]

K. Tamaki, M. Curty, G. Kato, H.-K. Lo, and K. Azuma, “Loss-tolerant quantum cryptography with imperfect sources,” Phys. Rev. A 90(5), 052314 (2014).
[Crossref]

H.-K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014).
[Crossref]

Debuisschert, T.

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
[Crossref]

Diamanti, E.

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
[Crossref]

Dianati, M.

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
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Ding, H.

J. Liu, H. Ding, C. Zhang, S. Xie, and Q. Wang, “Practical Phase-Modulation Stabilization in Quantum Key Distribution via Machine learning,” Phys. Rev. Appl. 12(1), 014059 (2019).
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Dixon, A. R.

Domeki, T.

Dynes, J. F.

G. L. Roberts, M. Pittaluga, M. Minder, M. Lucamarini, J. F. Dynes, Z. L. Yuan, and A. J. Shields, “Pattern-effect-free intensity modulator for secure decoy-state quantum key distribution,” Opt. Lett. 43(20), 5110–5113 (2018).
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M. Sasaki, M. Fujiwara, H. Ishizuka, W. Klaus, K. Wakui, M. Takeoka, S. Miki, T. Yamashita, Z. Wang, A. Tanaka, K. Yoshino, Y. Nambu, S. Takahashi, A. Tajima, A. Tomita, T. Domeki, T. Hasegawa, Y. Sakai, H. Kobayashi, T. Asai, K. Shimizu, T. Tokura, T. Tsurumaru, M. Matsui, T. Honjo, K. Tamaki, H. Takesue, Y. Tokura, J. F. Dynes, A. R. Dixon, A. W. Sharpe, Z. L. Yuan, A. J. Shields, S. Uchikoga, M. Legré, S. Robyr, P. Trinkler, L. Monat, J.-B. Page, G. Ribordy, A. Poppe, A. Allacher, O. Maurhart, T. Länger, M. Peev, and A. Zeilinger, “Field test of quantum key distribution in the Tokyo QKD network,” Opt. Express 19(11), 10387–10409 (2011).
[Crossref]

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
[Crossref]

A. R. Dixon, Z. L. Yuan, J. F. Dynes, A. W. Sharpe, and A. J. Shields, “Gigaherts decoy quantum key distribution with 1 Mbit/s secure key rate,” Opt. Express 16(23), 18790–18797 (2008).
[Crossref]

Fasel, S.

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
[Crossref]

Felber, N.

D. Stucki, M. Legré, F. Buntschu, B. Clausen, N. Felber, N. Gisin, L. Henzen, P. Junod, G. Litzistorf, P. Monbaron, L. Monat, J.-B. Page, D. Perroud, G. Ribordy, A. Rochas, S. Robyr, J. Tavares, R. Thew, P. Trinkler, S. Ventura, R. Voirol, N. Walenta, and H. Zbinden, “Long-term performance of the SwissQuantum quantum key distribution network in a field environment,” New J. Phys. 13(12), 123001 (2011).
[Crossref]

Filsinger, V.

T. Kawanishi, T. Sakamoto, A. Chiba, M. Izutsu, K. Higuma, J. Ichikawa, T. Lee, and V. Filsinger, “High-speed dual-parallel Mach-Zehnder modulator using thin lithium niobate substrate,” Proc. OFC/NFOEC, San Diego, CA, USA, 1–3 (2008).

Ford, C.

G. Bonfrate, M. Harlow, C. Ford, G. Maxwell, and P. D. Townsend, “Asymmetric Mach-Zehnder germane-silicate channel waveguide interferometers for quantum cryptography systems,” Electron. Lett. 37(13), 846–847 (2001).
[Crossref]

Fossier, S.

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
[Crossref]

Fujiwara, M.

Fürst, M.

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
[Crossref]

Gautier, J.-D.

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
[Crossref]

Gay, O.

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
[Crossref]

Gisin, N.

D. Stucki, M. Legré, F. Buntschu, B. Clausen, N. Felber, N. Gisin, L. Henzen, P. Junod, G. Litzistorf, P. Monbaron, L. Monat, J.-B. Page, D. Perroud, G. Ribordy, A. Rochas, S. Robyr, J. Tavares, R. Thew, P. Trinkler, S. Ventura, R. Voirol, N. Walenta, and H. Zbinden, “Long-term performance of the SwissQuantum quantum key distribution network in a field environment,” New J. Phys. 13(12), 123001 (2011).
[Crossref]

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
[Crossref]

Gottesman, D.

D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, “Security of quantum key distribution with imperfect devices,” Quantum Inf. Comput. 4(5), 135 (2004).
[Crossref]

Grangier, P.

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
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Robyr, S.

D. Stucki, M. Legré, F. Buntschu, B. Clausen, N. Felber, N. Gisin, L. Henzen, P. Junod, G. Litzistorf, P. Monbaron, L. Monat, J.-B. Page, D. Perroud, G. Ribordy, A. Rochas, S. Robyr, J. Tavares, R. Thew, P. Trinkler, S. Ventura, R. Voirol, N. Walenta, and H. Zbinden, “Long-term performance of the SwissQuantum quantum key distribution network in a field environment,” New J. Phys. 13(12), 123001 (2011).
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M. Sasaki, M. Fujiwara, H. Ishizuka, W. Klaus, K. Wakui, M. Takeoka, S. Miki, T. Yamashita, Z. Wang, A. Tanaka, K. Yoshino, Y. Nambu, S. Takahashi, A. Tajima, A. Tomita, T. Domeki, T. Hasegawa, Y. Sakai, H. Kobayashi, T. Asai, K. Shimizu, T. Tokura, T. Tsurumaru, M. Matsui, T. Honjo, K. Tamaki, H. Takesue, Y. Tokura, J. F. Dynes, A. R. Dixon, A. W. Sharpe, Z. L. Yuan, A. J. Shields, S. Uchikoga, M. Legré, S. Robyr, P. Trinkler, L. Monat, J.-B. Page, G. Ribordy, A. Poppe, A. Allacher, O. Maurhart, T. Länger, M. Peev, and A. Zeilinger, “Field test of quantum key distribution in the Tokyo QKD network,” Opt. Express 19(11), 10387–10409 (2011).
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D. Stucki, M. Legré, F. Buntschu, B. Clausen, N. Felber, N. Gisin, L. Henzen, P. Junod, G. Litzistorf, P. Monbaron, L. Monat, J.-B. Page, D. Perroud, G. Ribordy, A. Rochas, S. Robyr, J. Tavares, R. Thew, P. Trinkler, S. Ventura, R. Voirol, N. Walenta, and H. Zbinden, “Long-term performance of the SwissQuantum quantum key distribution network in a field environment,” New J. Phys. 13(12), 123001 (2011).
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Sasaki, T.

K. Yoshino, M. Fujiwara, K. Nakata, T. Sumiya, T. Sasaki, M. Takeoka, M. Sasaki, A. Tajima, M. Koashi, and A. Tomita, “Quantum key distribution with an efficient countermeasure against correlated intensity fluctuations in optical pulses,” Npj Quantum Inf. 4(1), 8 (2018).
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K. Yoshino, M. Fujiwara, K. Nakata, T. Sumiya, T. Sasaki, M. Takeoka, M. Sasaki, A. Tajima, M. Koashi, and A. Tomita, “Quantum key distribution with an efficient countermeasure against correlated intensity fluctuations in optical pulses,” Npj Quantum Inf. 4(1), 8 (2018).
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Takeoka, M.

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M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
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M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
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D. Stucki, M. Legré, F. Buntschu, B. Clausen, N. Felber, N. Gisin, L. Henzen, P. Junod, G. Litzistorf, P. Monbaron, L. Monat, J.-B. Page, D. Perroud, G. Ribordy, A. Rochas, S. Robyr, J. Tavares, R. Thew, P. Trinkler, S. Ventura, R. Voirol, N. Walenta, and H. Zbinden, “Long-term performance of the SwissQuantum quantum key distribution network in a field environment,” New J. Phys. 13(12), 123001 (2011).
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Figures (7)

Fig. 1.
Fig. 1. Typical implementation of phase-encoding BB84 QKD system (LD, laser diode; AMZI, asymmetric Mach–Zehnder interferometer; PM, phase modulator; ATT, attenuator; PD, photon detector). The inset shows a deformed square wave observed in a high-speed QKD system. A 20% fluctuation occurs in point A and B in square wave.
Fig. 2.
Fig. 2. Schematic of DDM ( ${{E}_\textrm{i}}$ , electric field of input light; ${{V}_\phi }$ , applied voltage to electrodes; $\phi $ , phase shift; ${{E}_{o}}$ , electric field of output light). DDM is based on MZI and modulates light intensity and phase by applying phase shifts independently on upper and lower paths.
Fig. 3.
Fig. 3. Schematic of DPM ( ${E_\textrm{i}}$ , electric field of input light; ${V_\phi }$ , applied voltage to electrodes; $\phi $ , phase shift; ${E_\textrm{o}}$ , electric field of output light; MZM, Mach–Zehnder modulator; i, $\pi /2$ phase shift). DPM is constructed with two push–pull Mach–Zehnder modulators connected in parallel.
Fig. 4.
Fig. 4. Experimental setup to investigate the effect of voltage shifts on phase-encoding BB84 QKD system (Iso, isolator; PC, polarization controller; PLC, planar light-wave circuit; PM, phase modulator; PPG, pulse pattern generator; ${V_\phi }$ , applied voltage to electrodes).
Fig. 5.
Fig. 5. Reconstructed state ${\rho _X}$ with DDM (a) without voltage variation and with (b) 25% DC bias and (c) 25% AC amplitude change. Reconstructed state ${\rho _X}$ with DPM (d) without voltage variation and with (e) 25% DC bias and (f) 25% AC amplitude change.
Fig. 6.
Fig. 6. (a) Fidelity according to DC bias. (b) Fidelity according to pulse amplitude (AC). Squares and stars represent the measured values for DDM and DPM, respectively. The solid and dotted lines represent the theoretical values for DDM and DPM, respectively.
Fig. 7.
Fig. 7. Final key generation rate by (a) DC and (b) AC variations in the modulator. Dashed lines represent voltage deviation of 0 (red), 0.05 ${{V}_{\pi}}$ (blue) and 0.1 ${{V}_{\pi}}$ (black) in DDM. Solid lines represent voltage deviation of 0 (red), 0.05 ${{V}_{\pi}}$ (blue), and 0.1 ${{V}_{\pi}}$ (black) and 0.2 ${{V}_{\pi}}$ (green)in DPM.

Tables (3)

Tables Icon

Table 1. Phase shifts in DDM to generate photon states X0, X1, Y0, and Y1. Variable $\phi _{i}^{({j} )}$ (i = 1, 2, j = 0, 1) denotes phase applied to i-th electrode for state $|{j}\rangle$ .

Tables Icon

Table 2. Phase shifts in DPM to generate photon states X0, X1, Y0, and Y1. Variable $\phi _{i}^{({j} )}$ (i = 1, 2, j = 0, 1) denotes phase applied to i-th electrode for state $|{j}\rangle$ .

Tables Icon

Table 3. Parameters for key generation rate

Equations (30)

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$$\Delta = \frac{{1 - {\mathop{F}\nolimits} ({{\rho_{\rm X}},{\rho_{\rm Y}}} )}}{2},$$
$${\mathop{F}\nolimits} ({{\rho_{\rm X}},{\rho_{\rm Y}}} ) = \textrm{tr}\left( {\sqrt {{{\left( {\sqrt {{\rho_{\rm X}}} \sqrt {{\rho_{\rm Y}}} } \right)}^\dagger }\sqrt {{\rho_{\rm X}}} \sqrt {{\rho_{\rm Y}}} } } \right).$$
$${\delta ^{\prime}_{\rm Y}} = {\delta _{\rm Y}} + 4\Delta + 4\sqrt {\Delta {\delta _{\rm Y}}}.$$
$$R \ge q[{{{Q}_1}({1 - H({{\delta^{\prime}_{\rm Y}} })} )- {Q_{\rm s}}fH({\delta_{\rm X}})} ],$$
$${E_o} = \frac{1}{2}({{e^{i{\phi_1}}} + {e^{i{\phi_2}}}} )= \cos \frac{{{\phi _1} - {\phi _2}}}{2}{e^{i\frac{{{\phi _1} + {\phi _2}}}{2}}}{E_i}.$$
$$|\Psi \rangle=\frac{1}{{\sqrt 2 }}\sum\limits_{j = 0,1} {\exp (} i\frac{{\phi _1^j + \phi _2^j}}{2})\cos \frac{{{\phi _1}^j - {\phi _2}^j}}{2}|j \rangle .$$
$$\begin{array}{c} {\textrm{exp}\left( {i\frac{{\phi_1^{(j )} + ({\phi_2^{(j )} + \varepsilon } )}}{2}} \right)\cos \frac{{\phi _1^{(j )} - ({\phi_2^{(j )} + \varepsilon } )}}{2}}\\ { = \textrm{exp}\left( {i\frac{{\phi_1^{(j )} + \phi_2^{(j )}}}{2}} \right)\left[ {({1 + i\varepsilon } )\cos \frac{{\phi_1^{(j )} - \phi_2^{(j )}}}{2} + \varepsilon \sin \frac{{\phi_1^{(j )} - \phi_2^{(j )}}}{2}} \right]} \end{array}$$
$${E_o} = \frac{{\cos {\phi _1} + i\cos {\phi _2}}}{{\sqrt 2 }}{E_i}.$$
$$|{\Psi}\rangle = \frac{1}{{\sqrt {\mathop \sum \nolimits_{j = 0.1} ({{{\cos }^2}{\phi_1}^{(j )} + {{\cos }^2}{\phi_2}^{(j )}} )} }}\mathop \sum \nolimits_{j = 0,1} ({\cos {\phi_1}^{(j )} + i\cos {\phi_2}^{(j )}} )|j\rangle.$$
$$\cos ({\phi _2} + \varepsilon ) = \cos {\phi _2}(1 - \frac{1}{{2!}}{\varepsilon ^2} + \frac{1}{{4!}}{\varepsilon ^4} - \cdots ),$$
$${\rho _\textrm{X}} = \frac{1}{2}(|{{\mathop{\rm X}\nolimits} 0^{\prime}} \rangle \left\langle {{\mathop{\rm X}\nolimits} 0^{\prime}} \right|+ |{{\mathop{\rm X}\nolimits} 1^{\prime}} \rangle \left\langle {{\mathop{\rm X}\nolimits} 1^{\prime}} \right|).$$
$$\rho = \frac{1}{2}\sum\limits_{i = 0}^3 {\frac{{{S_i}}}{{{S_0}}}{\sigma _i}} ,$$
$${S_0} = 2{n_0},$$
$${S_1} = 2({n_1} - {n_0}),$$
$${S_2} = 2({n_2} - {n_0}),$$
$${S_3} = 2({n_3} - {n_0}).$$
$${n_0} = \frac{N}{2}(\left\langle 0 \right|{\rho _X}|0 \rangle + \left\langle 1 \right|{\rho _X}|1 \rangle ) = \frac{N}{2}(P({{\mathop{\rm Z}\nolimits} 0|{\mathop{\rm X}\nolimits} 0^{\prime}} )+ P({{\mathop{\rm Z}\nolimits} 0|{\mathop{\rm X}\nolimits} 1^{\prime}} )+ P({{\mathop{\rm Z}\nolimits} 1|{\mathop{\rm X}\nolimits} 0^{\prime}} )+ P({{\mathop{\rm Z}\nolimits} 1|{\mathop{\rm X}\nolimits} 1^{\prime}} )),$$
$${n_1} = \frac{N}{2}(\left\langle 0 \right|{\rho _X}|0 \rangle ) = N(P({{\mathop{\rm Z}\nolimits} 0|{\mathop{\rm X}\nolimits} 0^{\prime}} )+ P({{\mathop{\rm Z}\nolimits} 0|{\mathop{\rm X}\nolimits} 1^{\prime}} )),$$
$${n_2} = \frac{N}{2}(\left\langle {{\mathop{\rm X}\nolimits} 1} \right|{\rho _X}|{{\mathop{\rm X}\nolimits} 1} \rangle ) = N(P({{\mathop{\rm X}\nolimits} 1|{\mathop{\rm X}\nolimits} 0^{\prime}} )+ P({{\mathop{\rm X}\nolimits} 1|{\mathop{\rm X}\nolimits} 1^{\prime}} )),$$
$${n_3} = \frac{N}{2}(\left\langle {{\mathop{\rm Y}\nolimits} 1} \right|{\rho _X}|{{\mathop{\rm Y}\nolimits} 1} \rangle = N(P({{\mathop{\rm Y}\nolimits} 1|{\mathop{\rm X}\nolimits} 0^{\prime}} )+ P({{\mathop{\rm Y}\nolimits} 1|{\mathop{\rm X}\nolimits} 1^{\prime}} )),$$
$$P({\mathop{\rm Z}\nolimits} 0|{\mathop{\rm X}\nolimits} 0^{\prime}) = \frac{{I({\mathop{\rm Z}\nolimits} 0|{\rm X}0^{\prime})}}{{I({\mathop{\rm Z}\nolimits} 0|{\rm X}0^{\prime}) + I({\mathop{\rm Z}\nolimits} 1|{\rm X}0^{\prime})}},$$
$${e_{i, j}} = \frac{1}{{{S_{{\mu _j}}}}}\left[ {\left( {1 - \exp \left( { - \eta {\mu_j}{{10}^{ - \frac{{\alpha l}}{{10}}}}} \right)} \right){E_i} + \frac{1}{2}\exp \left( { - \eta {\mu_j}{{10}^{ - \frac{{\alpha l}}{{10}}}}} \right){y_0}} \right], i = \textrm{X},\textrm{Y}, j = \textrm{d},\textrm{s}$$
$${S_{{\mu _\textrm{d}}}} = 1 + ( - 1 + {y_0})\exp \left( { - \eta {\mu_\textrm{d}}{{10}^{ - \frac{{\alpha l}}{{10}}}}} \right),$$
$${S_{{\mu _\textrm{s}}}} = 1 + ( - 1 + {y_0})\exp \left( { - \eta {\mu_\textrm{s}}{{10}^{ - \frac{{\alpha l}}{{10}}}}} \right).$$
$${E_i} = \frac{{1 - {v_i}}}{2}.$$
$${v_{\mathop{\rm X}\nolimits} } = \frac{{I({\mathop{\rm X}\nolimits} 0|{\rm X}0^{\prime}) - I({\mathop{\rm X}\nolimits} 1|{\rm X}0^{\prime})}}{{I({\mathop{\rm X}\nolimits} 0|{\rm X}0^{\prime}) + I({\mathop{\rm X}\nolimits} 1|{\rm X}0^{\prime})}},$$
$${v_{\mathop{\rm Y}\nolimits} } = \frac{{I({\mathop{\rm Y}\nolimits} 0|{\rm Y}0^{\prime}) - I({\mathop{\rm Y}\nolimits} 1|{\rm Y}0^{\prime})}}{{I({\mathop{\rm Y}\nolimits} 0|{\rm Y}0^{\prime}) + I({\mathop{\rm Y}\nolimits} 1|{\rm Y}0^{\prime})}}.$$
$${e_1} \le \frac{{{e_{{\rm Yd}}}{e^{{\mu _{\rm d}}}} - {e_0}{y_0}}}{{y_1^{L,{\mu _{\rm d}},0}{\mu _{\rm d}}}},$$
$$y_1^{L,{\mu _{\rm d}},0} = \frac{{{\mu _{\rm s}}}}{{{\mu _{\rm d}}{\mu _{\rm s}} - {\mu _{\rm d}}^2}}({Q_\textrm{d}}{e^{{\mu _{\rm s}}}} - {Q_\textrm{s}}{e^{{\mu _{\rm d}}}}\frac{{{\mu _{\rm d}}^2}}{{{\mu _{\rm s}}^2}} - \frac{{{\mu _{\rm s}}^2 - {\mu _{\rm d}}^2}}{{{\mu _{\rm s}}^2}}{y_0}),$$
$${Q_1} \ge Q_1^{L,{\mu _{\rm d}},0} = \frac{{\mu _{\rm s}^2}}{{{\mu _{\rm d}}{\mu _{\rm s}} - {\mu _{\rm d}}^2}}({S_{{\mu _{\rm d}}}}{e^{{\mu _{\rm s}}}} - {S_{{\mu _{\rm s}}}}{e^{{\mu _{\rm d}}}}\frac{{{\mu _{\rm d}}^2}}{{{\mu _{\rm s}}^2}} - \frac{{{\mu _{\rm s}}^2 - {\mu _{\rm d}}^2}}{{{\mu _{\rm s}}^2}}{y_0}),$$

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