Abstract

Excess noise induced by the phase drifts is a serious impairment for the continuous-variable quantum key distribution with locally generated local oscillator scheme, which is recently proposed to avoid the side channel attacks due to the transmitted local oscillator. Theoretical and experimental studies on the phase estimation have been widely reported, while two frequency-locked laser sources are indispensable to achieve quantum coherent detection. Moreover, the self-referenced phase estimation scheme requires to propagate the strong reference pulse through optical fiber, which opens a security loophole through the manipulation of the reference pulse amplitude. Based on the theoretical security and Bayes’ theorem, we propose a phase estimation protocol, which does not require propagating the strong reference pulse for performing phase estimation. Compared to the other related work, the protocol can avoid the security problem caused by strong reference pulse. Moreover, this algorithm is an iterative progress for each of experiment to obtain the phase estimation and its uncertainty. We hope the proposed scheme could further promote the performance of continuous-variable quantum key distribution.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. introduction

Quantum key distribution (QKD) [1–4], with the objective of sharing bits sequence between two legitimate partners, is the most advanced quantum technology to date. The first idea of QKD was the BB84 protocol [5]. Its key information is encoded on the polarization of the photon, and the specific components are employed to achieve single-photon detection. Unfortunately, imperfections of such devices can be exploited to attack system, namely the phase-remapping attack [6], the time-shift attack [7] and so forth. Protocol for an alternative approach, continuous-variable QKD (CVQKD) [8–10], takes the advantage of being compatible with standard telecommunication technology, especially no request on single-photon detectors. One well-known CVQKD protocol, in particular, is the Gaussian-modulated coherent-stated (GMCS) protocol [11]. For one thing, the GMCS-CVQKD protocol has the robustness against the incoherent background noise. On the other hand, the strong local oscillator (LO) can be deemed as a selective filter, which will availably suppress the noise photons in coherent detection.

As for the implementations of GMCS-CVQKD protocol, a major obstacle exists with this approach: Alice needs to prepare both the signal and LO from the same laser, and then propagate to Bob at the same time in order to reduce the phase noise [11,12]. Nevertheless, the implementation has several limitations. Firstly, the above implementation leaves a massive loophole open for eavesdroppers. Specially, eavesdroppers can manipulate the power of the LO to attack the system, such as intercept-resend attack [13], calibration attack [14], wavelength attack [15,16], where the LO is the phase reference required for coherent detection. Secondly, the efficiency of GMCS-CVQKD protocol will be reduced when sending the strong LO through optical fiber, meanwhile, the LO power will also be insufficient to operate well under the condition of shot-noise limit. The last but not the least, in the current fiber-based implementations, the technique of multiplexing needs to be adopted to transmit the LO with signal states. Hence, complicated de-multiplexing technique is necessary to separate the LO from the quantum signal at Bob’s side. Because of the photons leakage from the LO to weak quantum signal, the multiplexing scheme would cause intolerable “in-band” excess noise.

Fortunately, an improved GMCS-CVQKD protocol has been investigated recently, which can generate the LO “locally” by using an independent laser source at Bob’s side [17–19]. Considering that two independent laser sources exist issue of frequency instabilities, the performance and security may be reduced. Accordingly, establishing the phase reference and estimation between Alice and Bob becomes particularly crucial. Due to the weak quantum signal and low tolerable phase noise, traditional techniques (i.e. phase-locked loops [20] and feedforward carrier recovery [21]) are inappropriate in the coherent optical communication.

Considering the flaws of the GMCS-CVQKD protocol without sending a LO, several groups have investigated and gave solutions to the correlation technique. Daniel et al. displayed a CVQKD experiment by utilizing a delayed LO the at receiver’s side and compensating the phase drift with self-referenced scheme [19]. Qi et al. put forward a pilot-aided feedforward phase estimation scheme which performs coherent detection with balanced homodyne detectors [18], while another work of Huang et al. demonstrated a CVQKD experiment, which employing the LO with an extra laser source and a bandwidth shot-noise-limited homodyne detector at the receiver side [17]. Theoretical and experimental studies on the phase compensation have been widely reported, while two frequency-locked laser sources are indispensable to achieve quantum coherent detection. Furthermore, the techniques shown above are still employed strong referenced pulse to estimate and compensate phase. Recently, R. Shengjun et al. propose an attack that exploits the phase estimation error associated with the amplitude of the reference pulses [22]. They call this attack the “reference pulse attack”. Consequently, the security would be reduced when propagating the referenced pulse through the optical fiber.

Based on the theoretical security and Bayes’ theorem, we propose an efficient phase estimation protocol, which employs very weak laser with small mean photon count as the source. The protocol does not require propagating the strong reference pulse for performing phase estimation. Thus, compared to the other related work, the protocol can avoid the security problem caused by strong reference pulse. Bayesian phase estimation algorithm is introduced by Svore et al. [23]. First of all, an initial prior distribution is given to represent the confidence that the hypotheses are the correct eigenphase. Secondly, the experimental outcome is utilized to update the mean of phase and standard deviation according to the Bayesian theorem. Particularly, a host of particles are drawn from the prior distribution and then probabilistically discarded based on the likelihood function. Subsequently, the remaining samples model the posterior distribution, which becomes the new prior. In other words, this process is repeated for each of the random experiment in the dataset. According to the theoretical results, the Bayesian inference algorithm has several advantages. Firstly, it has the characteristic of high robustness to noise. Secondly, it employs the Bayesian estimator of the very weak laser source. Thus, it can avoid the security problem caused by the strong reference pulse. Thirdly, the algorithm can achieve a well-motivated confidence interval of the estimated eigenphase.

In this paper, we use the Bayesian inference algorithm to estimate the phase and its uncertainty for CVQKD protocol. Considering that large excess noise and low reconciliation efficiency are two drawbacks of CVQKD, we adopt the discrete modulation, namely the four-state CVQKD protocol to achieve a low signal-to-noise ratio and high reconciliation efficiency [24,25].

This paper is structured as follows. In Sec. 2, we describe the Bayesian inference algorithm for four-state CVQKD protocol, and then analyze the probability density function of phase shift and Bayesian posterior variance. In Sec. 3, we analyze the quantum bit error rate and give the model simulation results. Finally, the conclusion is drawn in Sec. 5.

2. Bayesian quantum inference algorithm

2.1. Four-state CVQKD protocol

In the general discrete-modulated CVQKD protocol, N coherent states |αk〉 = |αei(2k+1)π/N〉 with k ∈ {0, 1, 2, · · ·, N − 1} are prepared to carry information. Particularly, the four-state CVQKD protocol is considered as the fundamental communication protocol in our manuscript. At the transmitter (i.e., Alice), in order to send randomly one of the four coherent states, the encoding can be described as follows (see Fig. 1)

 

Fig. 1 The constellation of the four-state CVQKD scheme.

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Accordingly, for the four-state CVQKD protocol, we have |αk〉 = |αei(2k+1)π/4〉 with k ∈ {0, 1, 2, 3}, where α is a positive number related to the modulation variance V = 2α2, and the state |αi+1〉 is the phase shift of π/2 relative to the state |αi〉.

After Alice sends coherent states through the noisy quantum channel, the measurement of the phase shift should be performed with photon number resolving detector (PNRD) at the receiver, where the number of detected photons for four-state |αi〉 are denoted as ni with i ∈ {0, 1, 2, 3} respectively (as shown in Fig. 2). From a telecommunication point of view, the presence of a photon counting detector provides the possibility of generating a soft-metric at the receiver (as opposed to a hard-metric which essentially indicates the presence or an absence of a signal) that can be exploited in the informaiton reconciliation phase of the process. Consequently, the total number of detected photons at the receiver can be denoted as Nc=i=03ni. Subsequently, the Bayesian inference algorithm is applied to estimate the phase drift.

 

Fig. 2 The experiment step for the four-state CVQKD protocol. Alice sends coherent states through the noisy quantum channel, and then the measurement of phase shift should be performed with photon number resolving detector at the receiver. CW Laser, continuous-wave laser; AM, amplitude modulation; PM, phase modulation; PD, Photodiode; PNRD, photon number resolving detector.

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In what follows, the Bayesian phase estimation will be described in detail. Its main idea is described as follows. First of all, an initial prior distribution 𝒫(ϕ) is provided to express the confidence of the correct eigenphase in the current hypotheses. Subsequently, the measurement results of PNRD are utilized to update the initial prior probability distribution on the basis of Bayes’ rule. For instance, if the measurement outcome ∈ {|α0〉, |α1〉, |α2〉, |α3〉} is obtained by PNRD, so that the posterior probability distribution takes the following form [26]

𝒫(ϕ|)=𝒫(|ϕ)𝒫(ϕ)𝒫(|ϕ)𝒫(ϕ)dϕ.
Specifically, there are some particles drawn from the prior distribution, and then abandoned because that these particles does not match the likelihood function. The likelihood function for the four-state CVQKD protocol is defined following [27]
𝒫(|α0|ϕ)=14(1+eΔ2cos(ϕ)),𝒫(|α1|ϕ)=14(1+eΔ2sin(ϕ)),𝒫(|α2|ϕ)=14(1eΔ2cos(ϕ)),𝒫(|α3|ϕ)=14(1eΔ2sin(ϕ)).
It is assumed that during the propagation, the state undergoes a phase diffusion process whose amplitude is characterized by the parameter Δ. After updating the posterior distribution through Eq. (1), the posterior probability distribution becomes the new prior probability distribution. For each of the random simulation, the Bayesian phase estimation algorithm is an iterative process in the dataset.

2.2. Probability density function of phase shift

In the last section, the Bayesian phase estimation between prior probability distribution and posterior probability distribution has been detailed represented. In what follows, we define the probability density function (PDF) of phase shift and show the relation with the total number of detected photons 𝒩. The PDF of phase shift has the form as [27]

𝒫B(ϕ;𝒩)=i=03𝒫(|αi|ϕ)ni,
where is the normalization factor satisfies that
02π𝒫B(ϕ;𝒩)dϕ=1.

As can be seen in Fig. 3 and Fig. 4, we simulate the probability density function with different detected photons. On the basis of the encoding phase rule and likelihood function, the phase intervals [0, π) and [π, 2π) are the symmetrical interval. Therefore, the phase interval [0, π) is regarded as the case to analyze. As shown in Fig. 3, two shaded areas represent phase shift interval [0, π) and [π, 2π), respectively. From the first shaded area, we can see that the PDF of phase shift tends to 0 when the detected photons n0 increasing. Fig. 4 is the contour map of PDF with different detected photons. From the contour map, it is possible to observe that how the phase shift probability density function degree of concentration increased when the detected photons n0 increasing. Consequently, increasing the number of photons can improve the estimation accuracy of the four-state CVQKD protocol.

 

Fig. 3 The function of probability density function for phase drift. The detected photons at other three states are set as ni = 3 with i ∈ {1, 2, 3}, respectively.

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Fig. 4 The probability density function of phase shift at different detected photons n0. The detected photons are set to (a) n0 = 4, (b) n0 = 6, (c) n0 = 8 and (d) n0 = 10, respectively.

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2.3. Bayesian posterior variance

In this subsection, we analyze the mean phase of the distribution and Bayesian posterior variance [28–31]. In the Bayesian phase estimation algorithm, the measurement outcome is calculated with Bayes’ rule and likelihood function by a fixed but unknown ϕ′. In order to obtain the estimation of ϕ′, a parameter variable ϕ is introduced, so that the numerical estimate Φ() of ϕ′ can be made as [32]

Φ()=02πϕ𝒫(ϕ|)dϕ.

In what follows, we analyze the confidence interval of the estimator Φ(). Generally speaking, the variance of estimate Φ() can be described as [32]

(Δ2Φ())ϕ|=02π(ϕΦ())2𝒫(ϕ|)dϕ,
which is considered as the measurement of fluctuation about Φ(). Nevertheless, the outcome in Eq. (6) is an arbitrary measurement sequence. In other words, the definition of variance relies on the specific . Hence, averaging over the measurement sequences is necessary. Consequently, the average Bayesian posterior variance can be written as
(Δ2Φ),ϕ|ϕ=(Δ2Φ())ϕ|𝒫(|ϕ)=02π𝒫(ϕ|)𝒫(|ϕ)(ϕΦ())2dϕ,
where
𝒫(ϕ|)𝒫(|ϕ)=𝒫(ϕ,|ϕ).
Afterwards, we average over 𝒫(ϕ′) of the Bayesian posterior variance in Eq. (7), the expression can be described as
(Δ2Φ),ϕ,ϕ=02π(Δ2Φ),ϕ|ϕ𝒫(ϕ)dϕ=02π02π𝒫(ϕ|)𝒫(|ϕ)(ϕΦ())2𝒫(ϕ)dϕdϕ=02π𝒫(ϕ|)𝒫()(ϕΦ())2dϕ,
with the notation
𝒫()=02π𝒫(|ϕ)𝒫(ϕ)dϕ.
Considering the fluctuation of ϕ′, the probability 𝒫() is the average probability to observe the measurement sequence .

It is known that different statistical distribution of ϕ′ has the different prior probability 𝒫(ϕ). Specifically, in the case of the 𝒫(ϕ′) is known, the probability 𝒫(ϕ′) can be used as the prior probability in the Bayesian inference algorithm. In this condition, Eq. (9) can be simplified as

(Δ2Φ),ϕ=02π𝒫(,ϕ)(ϕΦ())2dϕ,
where
𝒫(,ϕ)=𝒫(ϕ|)𝒫().

3. Quantum bit error rate

In the section 2.1, the rule of encoding phase and likelihood function have been described and defined. In what follows, the transmission probability and transition probability will be expressed to estimate the quantum bit error rate (QBER) [33–35].

Considering the four-state CVQKD protocol, the encoding phase includes four kinds of conditions, namely ϕ0 = π/4, ϕ1 = 3π/4, ϕ2 = 5π/4 and ϕ3 = 7π/4, respectively. By further analysis of Eq. (2), the simplifications can be expressed as

𝒫(|α0|ϕ0)=14(1+eΔ2cos(π/4))=14(1+22eΔ2),𝒫(|α1|ϕ1)=14(1+eΔ2sin(3π/4))=14(1+22eΔ2),𝒫(|α2|ϕ2)=14(1+eΔ2cos(5π/4))=14(1+22eΔ2),𝒫(|α3|ϕ3)=14(1+eΔ2sin(7π/4))=14(1+22eΔ2).
It can be find that the four probability 𝒫(|α0〉|ϕ0), 𝒫(|α1〉|ϕ1), 𝒫(|α2〉|ϕ2) and 𝒫(|α3〉|ϕ3) are equivalent respectively. Subsequently, the transmission probability is defined with the following form [25]
𝒫ii=𝒫(|αi|ϕi),
with i ∈ {0, 1, 2, 3}, which satisfies the constraint
j=03𝒫(|αi|ϕj)=1.
According to the Eq. (14) and (15), the transition probability can be expressed as
𝒫ij=j=03𝒫(|αi|ϕj),
with ij, further simplification can be express as
𝒫ij=1𝒫ii.

Based on the photon counting detectors and Bayesian phase estimation algorithm, the QBER can be calculated as [25]

QBER=n𝒬nn,
with the notation
𝒬n={k=(n+1)/2nCnk𝒫ijk(1𝒫ij)nknisodd,k=(n+2)/2nCnk𝒫ijk(1𝒫ij)nk+12Cnn/2𝒫ijn/2(1𝒫ij)n/2niseven.
and
n=eNcNcnn!,Cnk=n!k!(nk)!n!=1×2×3×(n1)×n,
where n is the Poisson distribution because that the number of photons satisfies the poisson law [36–38], 𝒫ij has been defined in Eq. (16), Cnk and n ! represent the combination and factorial, respectively.

In the Eq. (19), n represents the transmitted photons per pulse and Nc represents the detected photons per pulse. If the channel is lossless, two parameters are satisfied n = Nc. If the channel is loss, two parameters are satisfied n > Nc. Comprehensively considering both the lossless channel and lossy channel, two parameters are subject to the restriction nNc. As shown in Fig. 5, the QBER is the function of transmitted photons n and detected photons Nc per pulse. Therefore, Fig. 5 is separated into feasible region (nNc) and infeasible region (n < Nc). For further analysis of the performance of QBER, Fig. 6 has been depicted. We can find that, increasing the number of detected photons per pulse Nc can reduce the QBER to an acceptable range.

 

Fig. 5 The mean quantum bit error rate (QBER) with four-state CVQKD protocol. Here, n represents the transmitted photons per pulse, and the detected photons per pulse are denoted as Nc. If the channel is lossless, two parameters are satisfied n = Nc. If the channel is loss, two parameters are satisfied n > Nc. Comprehensively considering both the lossless channel and lossy channel, two parameters are subject to the restriction nNc.

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Fig. 6 The mean quantum bit error rate (QBER). Here, Nc represents the detected photons per pulse, and the transmitted photons per pulse is set as n = 5.

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4. Model prediction

4.1. Inferences made with phase variance

In section 2.1, the main idea of Bayesian phase estimation algorithm has been described in detail. In what follows, we simulate the phase variance [26,39,40]. Firstly, we assume an initial prior distribution 𝒫(ϕ) over a suitable large range (μ, σ2) ∼ 𝒩(0, 103), here 𝒩 represents Normal distribution. Secondly, in order to obtain the experiment outcomes, we select 10 signal intensities (logarithmically spaced between 0.01 and 10). For each signal intensity, 100 simulated trials were performed. After performing a set of experimental measurements, the outcomes are used to update the mean of phase and its uncertainty σ based on Bayesian rule. The main steps are described in Tables 1, 2 and 3, and Fig. 7 is the simulation result for the posterior distribution of phase variance. It is well know that the posterior distribution is not exactly equal to the true value, not because of any error but because the estimation is based upon only 100 trials worth of data for each signal level.

Tables Icon

Table 1. Bayesian phase estimation algorithm and Bayesian phase estimation theory

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Table 2. Algorithm 1. Bayesian phase estimation.

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Table 3. Algorithm 2. Bayesian prior distribution updating function

 

Fig. 7 Inference made with the posterior distribution of phase variance. The prior distribution satisfies (μ, σ2) ∼ 𝒩(0, 103).

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Subsequently, in order to test whether the parameter estimate was sensitive to the different initial prior distribution, we simulate the procedure with different parameters, where the initial prior distribution 𝒩(μ, σ2) is set as (0, 10−3), (0, 10−2), (0, 10−1), (0, 100), (0, 101), (0, 102) and (0, 103), respectively. As shown in Fig. 8, the resulting parameter estimate is unaffected with different initial σ2. Consequently, the range of the initial prior distribution did not affect the parameter estimation procedure.

 

Fig. 8 The different parameter estimation procedure. Here, the parameter estimation procedure is repeated with the different initial prior distribution, such as 𝒩(μ, σ2) ∼ (0, 10−3), 𝒩(μ, σ2) ∼ (0, 10−2), 𝒩(μ, σ2) ∼ (0, 10−1), 𝒩(μ, σ2) ∼ (0, 1), 𝒩(μ, σ2) ∼ (0, 102) and 𝒩(μ, σ2) ∼ (0, 103), respectively.

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4.2. Model predicted performance

In what follows, we put forward the generative model. The model will be repeated in T trials, where T trials are repeated for each of C signal intensity levels. The C signal intensity levels can be denoted as

Δρ={Δρ1,Δρ2,Δρc}.
Therefore, there are C × T trials conducted. The model prediction consists of the following steps.
  • Step 1. On each trial t, the C signal intensity levels can be calculated as [41]
    Δρ=ρsρn,
    with
    ρs={ρs1,ρs2,,ρsc},
    and
    ρn={ρn1,ρn2,,ρnc},
    For each trial t, one signal item and one noise item will be displayed, with true feature values of ρsi and ρni, where ρs and ρn respresent the true feature values of signal item and phase noise item, respectively.
  • Step 2. The noisy sensory observation at the phase and signal locations can be denoted as xs and xn, respectively. Subsequently, we define the concept of correct response
    PC={PC1,PC2,,PCc}.
    On each trial t, the correct response are conducted when the signal and phase noise locations satisfies the constraint xs > xn. Hence, the correct response can be expressed as [41]
    PCi=𝒫(xs>xn)=Φ(Δρi2σ2),
    with i = 1, 2, · · ·, c, where Δρi represents the signal intensity levels, and function Φ(·) represents the cumulative standard normal distribution [42, 43]. Particularly, σ2 represents the phase variance, which has simulated in the section. 4.1 already.
  • Step 3. Based on the PC, the next step is to consider the actual proportion of correct response K/T and the relationship between PC and K/T for each of signal intensity conditions, where K = {k1, k2, · · · , kc} represents the number of correct responses PC in the T trials for each of the signal intensity level. Considering that the model prediction according to the Bernoulli trial, and ki is subject to Binomially distributed [44–46], so the parameter satisfy
    kiBinomial(PCi,T),
    with i = 1, 2, · · ·, c.

Subsequently, we use the parameter values σ2, PC and K to construct model and predict the distribution of correct responses. As shown in Fig. 9, we select 10 signal intensities, which is logarithmically spaced between 0.01 and 10 (the parameter C = 10). For each signal intensity, there are 100 simulated trials to perform (the parameter T = 100). Here, σ2 is treated as a constant for simplifying the calculation. According to the simulation result in Fig. 7, the parameter is set as σ2 = 1. Consequently, on the basis of Eq. (26) and (27), we can obtain the proportion correct K/T = {k1/T, k2/T, · · ·, kc/T}. As shown in the Fig. 9, the shaded region denotes the distribution of predicted number of correct trials. The simulation results show that Bayesian phase estimation can be utilized to effectively estimate the phase. Besides, the proportion correct (K/T) of concentration will be increased when the signal intensity level increasing.

 

Fig. 9 Inferences made with model prediction. The dataset (points) are utilized to estimate the posterior distribution of phase variance, which is analyzed in section 4.1. Subsequently, the posterior distribution of phase variance are utilized to generate the cumulative standard normal distribution of the model prediction.

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5. Conclusion

In this paper, we propose a phase estimation protocol for the CVQKD system. The protocol employs very weak laser source and Bayesian algorithm to estimate phase. Moreover, it does not require propagating the strong reference pulse in the quantum channel. However, the previous GMCS-CVQKD employed strong referenced pulse to estimate and compensate phase, which can be utilized by eavesdropper. Therefore, compared to the other related work, the protocol can avoid the security problem caused by strong reference pulse. Accordingly to the simulation model and results, the algorithm can achieve a well-motivated confidence interval of the estimated eigenphase. Consequently, the Bayesian phase estimation protocol can be applied in the CVQKD system.

Appendix

In the following, we derive the expression for the secret key rate. At the transmitter, Alice sends randomly one of the four coherent states to Bob through the channel. The total-added noise referred to the channel input is χline = 1/T′ + − 1, where T′ and represent the transmission efficiency and excess noise, respectively. The transmittance is defined as T′ = 10αL/10, where L is the fiber length and α is the loss coefficient of the optical fibers. At the receiver, Bob receives the coherent states and takes homodyne detection using a detector characterized by its electronics noise vel and efficiency η. The detection-added noise is defined as χhom = [(1 − η) + vel]/η. There, the total noise referred to the channel input can be expressed as χtot = χline + χhom/T.

When Alice and Bob use reverse reconciliation, the secret key rate can be defined as

K=βI(x:y)S(y:E),
where β is the reconciliation efficiency, I(x : y) is the nutual information between Alice and Bob, and S(y : E) is the mutual information between Bob and Eve. Specifically, the mutual information I(x : y) is given by [24]
I(x:y)=12log2V+χtot1+χtot,
with the notation V = VA + 1, where VA is the Alice’s modulation variance. The Holevo bound of the information between Eve and Bob is given by [25]
S(y:E)=G(λ112)+G(λ212)G(λ132)G(λ412),
where G(x) = (x + 1)log2(x + 1) − x log2x. The symplectic eigenvalues λ1,2 are given by
λ1,2=12(A±A24B,
where
A=V2+T2(V+χline22TZ42),B=(TV2+TVχlineTZ42)2.
Because that the covariance matrix has the same form as in the Gaussian modulation scheme, where Z4 can be replaced by the correlation of a two-mode squeezed vacuum ZG=VA2+2VA. The symplectic eigenvalues λ3,4 are given by [24,25]
λ3,4=12(C±C24D,
with the notation
C=Aχhom+VB+T(V+χline)T(V+χtot),D=BV+BχhomT(V+χtot).

Subsequently, we conduct numerical simulation using the realistic parameters. The parameters are summarized below: α = 0.2 dB/km, vel = 0.1, = 0.05, η = 0.5, β = 0.95 and VA = 0.3. Fig. 10 is the simulation result in the asymptotic scenario. Consequently, the proposed scheme can be applied to achieve efficient CVKQD system.

 

Fig. 10 The secret key rate for CVQKD protocol. Parameters are given as follows, α = 0.2 dB/km, vel = 0.1, = 0.05, η = 0.5, β = 0.95 and VA = 0.3.

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Funding

Fundamental Research Funds for the Central Universities of Central South University (2018zzts539); National Natural Science Foundation of China (NSFC) (61379153, 61572529).

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18. B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection,” Phys. Rev. X 5, 041009 (2015).

19. D. B. S. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R.M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5, 041010 (2015).

20. H.-C. Park, M. Lu, E. Bloch, T. Reed, Z. Griffith, L. Johansson, L. Coldren, and M. Rodwell, “40 Gbit/s Coherent Optical Receiver Using a Costas Loop,” Opt. Express 20, 197 (2012). [CrossRef]  

21. E. Ip and J. M. Kahn, “Feedforward Carrier Recovery for Coherent Optical Communications,” IEEE J. Lightw. Technol. 25, 2675 (2007). [CrossRef]  

22. S. Ren, R. Kumar, A. Wonfor, X. Tang, R. Penty, and I. White, “Reference pulse attack on continuous-variable quantum key distribution with local local oscillator,” (2017).

23. K. M. Svore, M. B. Hastings, and M. Freedman, “Faster phase estimation,” Quantum Information & Computation 14, 306 (2014).

24. H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A , 86, 22338 (2012). [CrossRef]  

25. B. Xu, C. Tang, H. Chen, W. Zhang, and F. Zhu, “Improve the maximum transmission distance of four-state continuous variable quantum key distribution by using a noiseless linear amplifier,” Phys. Rev. A , 378, 2808–2812 (2012).

26. N. Wiebe and C. Granade, “Efficient bayesian phase estimation,” Phys. Rev. Lett. 117, 010503 (2015). [CrossRef]  

27. F. Daneshgaran, M. T. Delgado, and M. Mondin, “Improved key rates for quantum key distribution employing soft metrics using Bayesian inference with photon counting detectors,” Quantum Communications and Quantum Imaging IX , 8163, 113–122 (2011).

28. M. J. W. Hall and H. M. Wiseman, “Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information,” New J. Phys. 14, 033040 (2012). [CrossRef]  

29. V. Giovannetti and L. Maccone, “Sub-heisenberg estimation strategies are ineffective,” Phys. Rev. Lett. 108, 210404 (2012). [CrossRef]  

30. L. Pezzè, P. Hyllus, and A. Smerzi, “Phase sensitivity bounds for two-mode interferometers,” Phys. Rev. A 91, 032103 (2015). [CrossRef]  

31. L. Pezzè, “Sub-Heisenberg phase uncertainties,” Phys. Rev. A 88, 060101 (2013). [CrossRef]  

32. Y. Li, L. Pezzè, M. Gessner, W. Li, and A. Smerzi, “Frequentist and bayesian quantum phase estimation,” Entropy 20, 628 (2018).

33. M. Nazarathy, E. Simony, and Y. Yadin, “Analytical evaluation of bit error rates for hard detection of optical differential phase amplitude shift keying (DPASK),” Journal of Lightwave Technology 24, 2248–2260 (2006). [CrossRef]  

34. M. A. Jing, Z. G. Yu, and Harbin, “Analysis of quantum bit error rate based on single-photon source with poisson distribution,” Optical Technique , 32, 101–104 (2006)

35. T. Ortlepp, H. Toepfer, and H. F. Uhlmann, “Minimization of noise-induced bit error rate in a high Tc superconducting dc/single flux quantum converter,” Applied Physics Letters 78, 1279–1281 (2001). [CrossRef]  

36. F. A. Haight, “Handbook of poisson distribution,” Journal of the Royal Statistical Society 18, 478 (1967).

37. P. C. Consul and G. C. Jain, “A Generalization of the Poisson Distribution,” Technometrics 15, 791–799 (1973). [CrossRef]  

38. E. G. Tsionas, “Bayesian analysis of the multivariate poisson distribution,” Communications in Statistics 28, 431–451 (1999). [CrossRef]  

39. S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. Obrien, and M. G. Thompson, “Experimental bayesian quantum phase estimation on a silicon photonic chip,” Phys. Rev. Lett. 118, 100503 (2017). [CrossRef]   [PubMed]  

40. N. Wiebe, C. Granade, C. Ferrie, and D. G. Cory, “Hamiltonian learning and certication using quantum resources,” Phys. Rev. Lett. 112, 190501 (2014). [CrossRef]  

41. B. T. Vincent, “A tutorial on Bayesian models of perception. Journal of Mathematical Psychology,” J. Math. Psychol. 66, 103–114 (2015). [CrossRef]  

42. P. N. Bowerman, R. G. Nolty, and E. M. Scheuer, “Calculation of the poisson cumulative distribution function (reliability applications),” IEEE Trans. Rel. 39, 158–161 (2002). [CrossRef]  

43. C. Dridi, “A short note on the numerical approximation of the standard normal cumulative distribution and its inverse,” Comput. Econ. 9, 77–81 (2003).

44. N. D Megill and M. Pavicic, “Estimating Bernoulli trial probability from a small sample,” https://arxiv.org/abs/1105.1486 (2011).

45. C. T. Le, “Analysis of nested designs with binomially-distributed outcome variables,” Biometrical J. , 29, 153–161 (2010). [CrossRef]  

46. F. J. Samaniego, “Maximum likelihood estimation for binomially distributed signals in discrete noise,” Publications of the American Statistical Association 75, 117–121 (1980). [CrossRef]  

References

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  1. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The Security of Practical Quantum Key Distribution,” Rev. Mod. Phys. 81, 1301 (2009).
    [Crossref]
  2. H. K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2015).
    [Crossref]
  3. L. B. Samuel and V. L. Peter, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
    [Crossref]
  4. C. Weedbrook, S. Pirandola, R. Garcá-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
    [Crossref]
  5. S. K. Chong and T. H. wang, “Quantum key agreement protocol based on BB84,” Opt. Commun. 283(6), 1192–1195 (2010).
    [Crossref]
  6. Y. Zhao, C. H. F. Fung, B. Qi, C. Chen, and H. Lo, “Quantum hacking : Experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Physical Review A 78(4). 042333 (2008).
    [Crossref]
  7. F. Xu, B. Qi, and H. K. Lo, “Experimental demonstration of phase-remapping attack in a practical quantum key distribution system,” New J. Phys 12(11), 63–66 (2010).
    [Crossref]
  8. F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88(5), 057902 (2002).
    [Crossref] [PubMed]
  9. A. M. Lance, T. Symul, and V. Sharma, “No-switching quantum key distribution using broadband modulated coherent light,” Phys. Rev. Lett. 95(18), 180503 (2005).
    [Crossref] [PubMed]
  10. Raúl García-Patrón and Nicolas J. Cerf, “Unconditional optimality of Gaussian attacks against continuous-variable quantum key distribution,” Phys. Rev. Lett. 97 (19), 190503 (2006).
    [Crossref] [PubMed]
  11. F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
    [Crossref] [PubMed]
  12. P. Jouguet, S. Kunz-Jacques, A. Leverrier, Ph. Grangier, and E. Diamanti, “Experimental Demonstration of Long-Distance Continuous-Variable Quantum Key Distribution,” Nat. Photonics 7, 378 (2013).
    [Crossref]
  13. J. Lin, H. Y. Tseng, and T. Hwang, “Intercept-resend attacks on Chen et al.’s quantum private comparison protocol and the improvements,” Opt. Commun. 284(9), 2412–2414 (2011).
    [Crossref]
  14. P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys Rev. A 87, 062313 (2013).
    [Crossref]
  15. J. Z. Huang, C. Weedbrook, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, G. C. Guo, and Z. F. Han, “Quantum hacking on continuous-variable quantum key distribution system using a wavelength attack,” Phys. Rev.A 87, 062329 (2013).
    [Crossref]
  16. X. C. Ma, S. H. Sun, M. S. Jiang, and L. M. Liang, “Wavelength attack on practical continuous-variable quantum-key-distribution system with a heterodyne protocol,” Phys. Rev. A 87, 052309 (2013).
    [Crossref]
  17. D. Huang, P. Huang, D. Lin, C. Wang, and G. Zeng, “High-speed continuous-variable quantum key distribution without sending a local oscillator,” Opt. Lett. 40(16), 3695–3698 (2015).
    [Crossref] [PubMed]
  18. B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection,” Phys. Rev. X 5, 041009 (2015).
  19. D. B. S. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R.M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5, 041010 (2015).
  20. H.-C. Park, M. Lu, E. Bloch, T. Reed, Z. Griffith, L. Johansson, L. Coldren, and M. Rodwell, “40 Gbit/s Coherent Optical Receiver Using a Costas Loop,” Opt. Express 20, 197 (2012).
    [Crossref]
  21. E. Ip and J. M. Kahn, “Feedforward Carrier Recovery for Coherent Optical Communications,” IEEE J. Lightw. Technol. 25, 2675 (2007).
    [Crossref]
  22. S. Ren, R. Kumar, A. Wonfor, X. Tang, R. Penty, and I. White, “Reference pulse attack on continuous-variable quantum key distribution with local local oscillator,” (2017).
  23. K. M. Svore, M. B. Hastings, and M. Freedman, “Faster phase estimation,” Quantum Information & Computation 14, 306 (2014).
  24. H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A,  86, 22338 (2012).
    [Crossref]
  25. B. Xu, C. Tang, H. Chen, W. Zhang, and F. Zhu, “Improve the maximum transmission distance of four-state continuous variable quantum key distribution by using a noiseless linear amplifier,” Phys. Rev. A,  378, 2808–2812 (2012).
  26. N. Wiebe and C. Granade, “Efficient bayesian phase estimation,” Phys. Rev. Lett. 117, 010503 (2015).
    [Crossref]
  27. F. Daneshgaran, M. T. Delgado, and M. Mondin, “Improved key rates for quantum key distribution employing soft metrics using Bayesian inference with photon counting detectors,” Quantum Communications and Quantum Imaging IX,  8163, 113–122 (2011).
  28. M. J. W. Hall and H. M. Wiseman, “Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information,” New J. Phys. 14, 033040 (2012).
    [Crossref]
  29. V. Giovannetti and L. Maccone, “Sub-heisenberg estimation strategies are ineffective,” Phys. Rev. Lett. 108, 210404 (2012).
    [Crossref]
  30. L. Pezzè, P. Hyllus, and A. Smerzi, “Phase sensitivity bounds for two-mode interferometers,” Phys. Rev. A 91, 032103 (2015).
    [Crossref]
  31. L. Pezzè, “Sub-Heisenberg phase uncertainties,” Phys. Rev. A 88, 060101 (2013).
    [Crossref]
  32. Y. Li, L. Pezzè, M. Gessner, W. Li, and A. Smerzi, “Frequentist and bayesian quantum phase estimation,” Entropy 20, 628 (2018).
  33. M. Nazarathy, E. Simony, and Y. Yadin, “Analytical evaluation of bit error rates for hard detection of optical differential phase amplitude shift keying (DPASK),” Journal of Lightwave Technology 24, 2248–2260 (2006).
    [Crossref]
  34. M. A. Jing, Z. G. Yu, and Harbin, “Analysis of quantum bit error rate based on single-photon source with poisson distribution,” Optical Technique,  32, 101–104 (2006)
  35. T. Ortlepp, H. Toepfer, and H. F. Uhlmann, “Minimization of noise-induced bit error rate in a high Tc superconducting dc/single flux quantum converter,” Applied Physics Letters 78, 1279–1281 (2001).
    [Crossref]
  36. F. A. Haight, “Handbook of poisson distribution,” Journal of the Royal Statistical Society 18, 478 (1967).
  37. P. C. Consul and G. C. Jain, “A Generalization of the Poisson Distribution,” Technometrics 15, 791–799 (1973).
    [Crossref]
  38. E. G. Tsionas, “Bayesian analysis of the multivariate poisson distribution,” Communications in Statistics 28, 431–451 (1999).
    [Crossref]
  39. S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. Obrien, and M. G. Thompson, “Experimental bayesian quantum phase estimation on a silicon photonic chip,” Phys. Rev. Lett. 118, 100503 (2017).
    [Crossref] [PubMed]
  40. N. Wiebe, C. Granade, C. Ferrie, and D. G. Cory, “Hamiltonian learning and certication using quantum resources,” Phys. Rev. Lett. 112, 190501 (2014).
    [Crossref]
  41. B. T. Vincent, “A tutorial on Bayesian models of perception. Journal of Mathematical Psychology,” J. Math. Psychol. 66, 103–114 (2015).
    [Crossref]
  42. P. N. Bowerman, R. G. Nolty, and E. M. Scheuer, “Calculation of the poisson cumulative distribution function (reliability applications),” IEEE Trans. Rel. 39, 158–161 (2002).
    [Crossref]
  43. C. Dridi, “A short note on the numerical approximation of the standard normal cumulative distribution and its inverse,” Comput. Econ. 9, 77–81 (2003).
  44. N. D Megill and M. Pavicic, “Estimating Bernoulli trial probability from a small sample,” https://arxiv.org/abs/1105.1486 (2011).
  45. C. T. Le, “Analysis of nested designs with binomially-distributed outcome variables,” Biometrical J.,  29, 153–161 (2010).
    [Crossref]
  46. F. J. Samaniego, “Maximum likelihood estimation for binomially distributed signals in discrete noise,” Publications of the American Statistical Association 75, 117–121 (1980).
    [Crossref]

2018 (1)

Y. Li, L. Pezzè, M. Gessner, W. Li, and A. Smerzi, “Frequentist and bayesian quantum phase estimation,” Entropy 20, 628 (2018).

2017 (1)

S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. Obrien, and M. G. Thompson, “Experimental bayesian quantum phase estimation on a silicon photonic chip,” Phys. Rev. Lett. 118, 100503 (2017).
[Crossref] [PubMed]

2015 (7)

B. T. Vincent, “A tutorial on Bayesian models of perception. Journal of Mathematical Psychology,” J. Math. Psychol. 66, 103–114 (2015).
[Crossref]

L. Pezzè, P. Hyllus, and A. Smerzi, “Phase sensitivity bounds for two-mode interferometers,” Phys. Rev. A 91, 032103 (2015).
[Crossref]

N. Wiebe and C. Granade, “Efficient bayesian phase estimation,” Phys. Rev. Lett. 117, 010503 (2015).
[Crossref]

H. K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2015).
[Crossref]

D. Huang, P. Huang, D. Lin, C. Wang, and G. Zeng, “High-speed continuous-variable quantum key distribution without sending a local oscillator,” Opt. Lett. 40(16), 3695–3698 (2015).
[Crossref] [PubMed]

B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection,” Phys. Rev. X 5, 041009 (2015).

D. B. S. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R.M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5, 041010 (2015).

2014 (2)

K. M. Svore, M. B. Hastings, and M. Freedman, “Faster phase estimation,” Quantum Information & Computation 14, 306 (2014).

N. Wiebe, C. Granade, C. Ferrie, and D. G. Cory, “Hamiltonian learning and certication using quantum resources,” Phys. Rev. Lett. 112, 190501 (2014).
[Crossref]

2013 (5)

P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys Rev. A 87, 062313 (2013).
[Crossref]

J. Z. Huang, C. Weedbrook, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, G. C. Guo, and Z. F. Han, “Quantum hacking on continuous-variable quantum key distribution system using a wavelength attack,” Phys. Rev.A 87, 062329 (2013).
[Crossref]

X. C. Ma, S. H. Sun, M. S. Jiang, and L. M. Liang, “Wavelength attack on practical continuous-variable quantum-key-distribution system with a heterodyne protocol,” Phys. Rev. A 87, 052309 (2013).
[Crossref]

L. Pezzè, “Sub-Heisenberg phase uncertainties,” Phys. Rev. A 88, 060101 (2013).
[Crossref]

P. Jouguet, S. Kunz-Jacques, A. Leverrier, Ph. Grangier, and E. Diamanti, “Experimental Demonstration of Long-Distance Continuous-Variable Quantum Key Distribution,” Nat. Photonics 7, 378 (2013).
[Crossref]

2012 (6)

H.-C. Park, M. Lu, E. Bloch, T. Reed, Z. Griffith, L. Johansson, L. Coldren, and M. Rodwell, “40 Gbit/s Coherent Optical Receiver Using a Costas Loop,” Opt. Express 20, 197 (2012).
[Crossref]

C. Weedbrook, S. Pirandola, R. Garcá-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A,  86, 22338 (2012).
[Crossref]

B. Xu, C. Tang, H. Chen, W. Zhang, and F. Zhu, “Improve the maximum transmission distance of four-state continuous variable quantum key distribution by using a noiseless linear amplifier,” Phys. Rev. A,  378, 2808–2812 (2012).

M. J. W. Hall and H. M. Wiseman, “Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information,” New J. Phys. 14, 033040 (2012).
[Crossref]

V. Giovannetti and L. Maccone, “Sub-heisenberg estimation strategies are ineffective,” Phys. Rev. Lett. 108, 210404 (2012).
[Crossref]

2011 (2)

F. Daneshgaran, M. T. Delgado, and M. Mondin, “Improved key rates for quantum key distribution employing soft metrics using Bayesian inference with photon counting detectors,” Quantum Communications and Quantum Imaging IX,  8163, 113–122 (2011).

J. Lin, H. Y. Tseng, and T. Hwang, “Intercept-resend attacks on Chen et al.’s quantum private comparison protocol and the improvements,” Opt. Commun. 284(9), 2412–2414 (2011).
[Crossref]

2010 (3)

S. K. Chong and T. H. wang, “Quantum key agreement protocol based on BB84,” Opt. Commun. 283(6), 1192–1195 (2010).
[Crossref]

F. Xu, B. Qi, and H. K. Lo, “Experimental demonstration of phase-remapping attack in a practical quantum key distribution system,” New J. Phys 12(11), 63–66 (2010).
[Crossref]

C. T. Le, “Analysis of nested designs with binomially-distributed outcome variables,” Biometrical J.,  29, 153–161 (2010).
[Crossref]

2009 (1)

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The Security of Practical Quantum Key Distribution,” Rev. Mod. Phys. 81, 1301 (2009).
[Crossref]

2008 (1)

Y. Zhao, C. H. F. Fung, B. Qi, C. Chen, and H. Lo, “Quantum hacking : Experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Physical Review A 78(4). 042333 (2008).
[Crossref]

2007 (1)

E. Ip and J. M. Kahn, “Feedforward Carrier Recovery for Coherent Optical Communications,” IEEE J. Lightw. Technol. 25, 2675 (2007).
[Crossref]

2006 (3)

Raúl García-Patrón and Nicolas J. Cerf, “Unconditional optimality of Gaussian attacks against continuous-variable quantum key distribution,” Phys. Rev. Lett. 97 (19), 190503 (2006).
[Crossref] [PubMed]

M. Nazarathy, E. Simony, and Y. Yadin, “Analytical evaluation of bit error rates for hard detection of optical differential phase amplitude shift keying (DPASK),” Journal of Lightwave Technology 24, 2248–2260 (2006).
[Crossref]

M. A. Jing, Z. G. Yu, and Harbin, “Analysis of quantum bit error rate based on single-photon source with poisson distribution,” Optical Technique,  32, 101–104 (2006)

2005 (2)

L. B. Samuel and V. L. Peter, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

A. M. Lance, T. Symul, and V. Sharma, “No-switching quantum key distribution using broadband modulated coherent light,” Phys. Rev. Lett. 95(18), 180503 (2005).
[Crossref] [PubMed]

2003 (2)

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[Crossref] [PubMed]

C. Dridi, “A short note on the numerical approximation of the standard normal cumulative distribution and its inverse,” Comput. Econ. 9, 77–81 (2003).

2002 (2)

P. N. Bowerman, R. G. Nolty, and E. M. Scheuer, “Calculation of the poisson cumulative distribution function (reliability applications),” IEEE Trans. Rel. 39, 158–161 (2002).
[Crossref]

F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88(5), 057902 (2002).
[Crossref] [PubMed]

2001 (1)

T. Ortlepp, H. Toepfer, and H. F. Uhlmann, “Minimization of noise-induced bit error rate in a high Tc superconducting dc/single flux quantum converter,” Applied Physics Letters 78, 1279–1281 (2001).
[Crossref]

1999 (1)

E. G. Tsionas, “Bayesian analysis of the multivariate poisson distribution,” Communications in Statistics 28, 431–451 (1999).
[Crossref]

1980 (1)

F. J. Samaniego, “Maximum likelihood estimation for binomially distributed signals in discrete noise,” Publications of the American Statistical Association 75, 117–121 (1980).
[Crossref]

1973 (1)

P. C. Consul and G. C. Jain, “A Generalization of the Poisson Distribution,” Technometrics 15, 791–799 (1973).
[Crossref]

1967 (1)

F. A. Haight, “Handbook of poisson distribution,” Journal of the Royal Statistical Society 18, 478 (1967).

Assche, G. V.

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[Crossref] [PubMed]

Bechmann-Pasquinucci, H.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The Security of Practical Quantum Key Distribution,” Rev. Mod. Phys. 81, 1301 (2009).
[Crossref]

Bloch, E.

H.-C. Park, M. Lu, E. Bloch, T. Reed, Z. Griffith, L. Johansson, L. Coldren, and M. Rodwell, “40 Gbit/s Coherent Optical Receiver Using a Costas Loop,” Opt. Express 20, 197 (2012).
[Crossref]

Bobrek, M.

B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection,” Phys. Rev. X 5, 041009 (2015).

Bowerman, P. N.

P. N. Bowerman, R. G. Nolty, and E. M. Scheuer, “Calculation of the poisson cumulative distribution function (reliability applications),” IEEE Trans. Rel. 39, 158–161 (2002).
[Crossref]

Brif, C.

D. B. S. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R.M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5, 041010 (2015).

Brouri, R.

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[Crossref] [PubMed]

Camacho, R.M.

D. B. S. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R.M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5, 041010 (2015).

Cerf, N. J.

C. Weedbrook, S. Pirandola, R. Garcá-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The Security of Practical Quantum Key Distribution,” Rev. Mod. Phys. 81, 1301 (2009).
[Crossref]

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[Crossref] [PubMed]

Cerf, Nicolas J.

Raúl García-Patrón and Nicolas J. Cerf, “Unconditional optimality of Gaussian attacks against continuous-variable quantum key distribution,” Phys. Rev. Lett. 97 (19), 190503 (2006).
[Crossref] [PubMed]

Chen, C.

Y. Zhao, C. H. F. Fung, B. Qi, C. Chen, and H. Lo, “Quantum hacking : Experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Physical Review A 78(4). 042333 (2008).
[Crossref]

Chen, H.

B. Xu, C. Tang, H. Chen, W. Zhang, and F. Zhu, “Improve the maximum transmission distance of four-state continuous variable quantum key distribution by using a noiseless linear amplifier,” Phys. Rev. A,  378, 2808–2812 (2012).

Chen, W.

J. Z. Huang, C. Weedbrook, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, G. C. Guo, and Z. F. Han, “Quantum hacking on continuous-variable quantum key distribution system using a wavelength attack,” Phys. Rev.A 87, 062329 (2013).
[Crossref]

Chong, S. K.

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Y. Li, L. Pezzè, M. Gessner, W. Li, and A. Smerzi, “Frequentist and bayesian quantum phase estimation,” Entropy 20, 628 (2018).

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X. C. Ma, S. H. Sun, M. S. Jiang, and L. M. Liang, “Wavelength attack on practical continuous-variable quantum-key-distribution system with a heterodyne protocol,” Phys. Rev. A 87, 052309 (2013).
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Wang, S.

J. Z. Huang, C. Weedbrook, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, G. C. Guo, and Z. F. Han, “Quantum hacking on continuous-variable quantum key distribution system using a wavelength attack,” Phys. Rev.A 87, 062329 (2013).
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S. K. Chong and T. H. wang, “Quantum key agreement protocol based on BB84,” Opt. Commun. 283(6), 1192–1195 (2010).
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J. Z. Huang, C. Weedbrook, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, G. C. Guo, and Z. F. Han, “Quantum hacking on continuous-variable quantum key distribution system using a wavelength attack,” Phys. Rev.A 87, 062329 (2013).
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C. Weedbrook, S. Pirandola, R. Garcá-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
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Wiebe, N.

S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. Obrien, and M. G. Thompson, “Experimental bayesian quantum phase estimation on a silicon photonic chip,” Phys. Rev. Lett. 118, 100503 (2017).
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N. Wiebe, C. Granade, C. Ferrie, and D. G. Cory, “Hamiltonian learning and certication using quantum resources,” Phys. Rev. Lett. 112, 190501 (2014).
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M. J. W. Hall and H. M. Wiseman, “Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information,” New J. Phys. 14, 033040 (2012).
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M. Nazarathy, E. Simony, and Y. Yadin, “Analytical evaluation of bit error rates for hard detection of optical differential phase amplitude shift keying (DPASK),” Journal of Lightwave Technology 24, 2248–2260 (2006).
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B. Xu, C. Tang, H. Chen, W. Zhang, and F. Zhu, “Improve the maximum transmission distance of four-state continuous variable quantum key distribution by using a noiseless linear amplifier,” Phys. Rev. A,  378, 2808–2812 (2012).

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B. Xu, C. Tang, H. Chen, W. Zhang, and F. Zhu, “Improve the maximum transmission distance of four-state continuous variable quantum key distribution by using a noiseless linear amplifier,” Phys. Rev. A,  378, 2808–2812 (2012).

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M. Nazarathy, E. Simony, and Y. Yadin, “Analytical evaluation of bit error rates for hard detection of optical differential phase amplitude shift keying (DPASK),” Journal of Lightwave Technology 24, 2248–2260 (2006).
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Nature (1)

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[Crossref] [PubMed]

New J. Phys (1)

F. Xu, B. Qi, and H. K. Lo, “Experimental demonstration of phase-remapping attack in a practical quantum key distribution system,” New J. Phys 12(11), 63–66 (2010).
[Crossref]

New J. Phys. (1)

M. J. W. Hall and H. M. Wiseman, “Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information,” New J. Phys. 14, 033040 (2012).
[Crossref]

Opt. Commun. (2)

S. K. Chong and T. H. wang, “Quantum key agreement protocol based on BB84,” Opt. Commun. 283(6), 1192–1195 (2010).
[Crossref]

J. Lin, H. Y. Tseng, and T. Hwang, “Intercept-resend attacks on Chen et al.’s quantum private comparison protocol and the improvements,” Opt. Commun. 284(9), 2412–2414 (2011).
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Opt. Lett. (1)

Optical Technique (1)

M. A. Jing, Z. G. Yu, and Harbin, “Analysis of quantum bit error rate based on single-photon source with poisson distribution,” Optical Technique,  32, 101–104 (2006)

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P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys Rev. A 87, 062313 (2013).
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[Crossref]

H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A,  86, 22338 (2012).
[Crossref]

B. Xu, C. Tang, H. Chen, W. Zhang, and F. Zhu, “Improve the maximum transmission distance of four-state continuous variable quantum key distribution by using a noiseless linear amplifier,” Phys. Rev. A,  378, 2808–2812 (2012).

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N. Wiebe and C. Granade, “Efficient bayesian phase estimation,” Phys. Rev. Lett. 117, 010503 (2015).
[Crossref]

V. Giovannetti and L. Maccone, “Sub-heisenberg estimation strategies are ineffective,” Phys. Rev. Lett. 108, 210404 (2012).
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[Crossref] [PubMed]

S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. Obrien, and M. G. Thompson, “Experimental bayesian quantum phase estimation on a silicon photonic chip,” Phys. Rev. Lett. 118, 100503 (2017).
[Crossref] [PubMed]

N. Wiebe, C. Granade, C. Ferrie, and D. G. Cory, “Hamiltonian learning and certication using quantum resources,” Phys. Rev. Lett. 112, 190501 (2014).
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B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection,” Phys. Rev. X 5, 041009 (2015).

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Phys. Rev.A (1)

J. Z. Huang, C. Weedbrook, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, G. C. Guo, and Z. F. Han, “Quantum hacking on continuous-variable quantum key distribution system using a wavelength attack,” Phys. Rev.A 87, 062329 (2013).
[Crossref]

Physical Review A (1)

Y. Zhao, C. H. F. Fung, B. Qi, C. Chen, and H. Lo, “Quantum hacking : Experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Physical Review A 78(4). 042333 (2008).
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S. Ren, R. Kumar, A. Wonfor, X. Tang, R. Penty, and I. White, “Reference pulse attack on continuous-variable quantum key distribution with local local oscillator,” (2017).

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Figures (10)

Fig. 1
Fig. 1 The constellation of the four-state CVQKD scheme.
Fig. 2
Fig. 2 The experiment step for the four-state CVQKD protocol. Alice sends coherent states through the noisy quantum channel, and then the measurement of phase shift should be performed with photon number resolving detector at the receiver. CW Laser, continuous-wave laser; AM, amplitude modulation; PM, phase modulation; PD, Photodiode; PNRD, photon number resolving detector.
Fig. 3
Fig. 3 The function of probability density function for phase drift. The detected photons at other three states are set as ni = 3 with i ∈ {1, 2, 3}, respectively.
Fig. 4
Fig. 4 The probability density function of phase shift at different detected photons n0. The detected photons are set to (a) n0 = 4, (b) n0 = 6, (c) n0 = 8 and (d) n0 = 10, respectively.
Fig. 5
Fig. 5 The mean quantum bit error rate (QBER) with four-state CVQKD protocol. Here, n represents the transmitted photons per pulse, and the detected photons per pulse are denoted as Nc. If the channel is lossless, two parameters are satisfied n = Nc. If the channel is loss, two parameters are satisfied n > Nc. Comprehensively considering both the lossless channel and lossy channel, two parameters are subject to the restriction nNc.
Fig. 6
Fig. 6 The mean quantum bit error rate (QBER). Here, Nc represents the detected photons per pulse, and the transmitted photons per pulse is set as n = 5.
Fig. 7
Fig. 7 Inference made with the posterior distribution of phase variance. The prior distribution satisfies (μ, σ2) ∼ ��(0, 103).
Fig. 8
Fig. 8 The different parameter estimation procedure. Here, the parameter estimation procedure is repeated with the different initial prior distribution, such as ��(μ, σ2) ∼ (0, 10−3), ��(μ, σ2) ∼ (0, 10−2), ��(μ, σ2) ∼ (0, 10−1), ��(μ, σ2) ∼ (0, 1), ��(μ, σ2) ∼ (0, 102) and ��(μ, σ2) ∼ (0, 103), respectively.
Fig. 9
Fig. 9 Inferences made with model prediction. The dataset (points) are utilized to estimate the posterior distribution of phase variance, which is analyzed in section 4.1. Subsequently, the posterior distribution of phase variance are utilized to generate the cumulative standard normal distribution of the model prediction.
Fig. 10
Fig. 10 The secret key rate for CVQKD protocol. Parameters are given as follows, α = 0.2 dB/km, vel = 0.1, = 0.05, η = 0.5, β = 0.95 and VA = 0.3.

Tables (3)

Tables Icon

Table 1 Bayesian phase estimation algorithm and Bayesian phase estimation theory

Tables Icon

Table 2 Algorithm 1. Bayesian phase estimation.

Tables Icon

Table 3 Algorithm 2. Bayesian prior distribution updating function

Equations (34)

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𝒫 ( ϕ | ) = 𝒫 ( | ϕ ) 𝒫 ( ϕ ) 𝒫 ( | ϕ ) 𝒫 ( ϕ ) d ϕ .
𝒫 ( | α 0 | ϕ ) = 1 4 ( 1 + e Δ 2 cos ( ϕ ) ) , 𝒫 ( | α 1 | ϕ ) = 1 4 ( 1 + e Δ 2 sin ( ϕ ) ) , 𝒫 ( | α 2 | ϕ ) = 1 4 ( 1 e Δ 2 cos ( ϕ ) ) , 𝒫 ( | α 3 | ϕ ) = 1 4 ( 1 e Δ 2 sin ( ϕ ) ) .
𝒫 B ( ϕ ; 𝒩 ) = i = 0 3 𝒫 ( | α i | ϕ ) n i ,
0 2 π 𝒫 B ( ϕ ; 𝒩 ) d ϕ = 1 .
Φ ( ) = 0 2 π ϕ 𝒫 ( ϕ | ) d ϕ .
( Δ 2 Φ ( ) ) ϕ | = 0 2 π ( ϕ Φ ( ) ) 2 𝒫 ( ϕ | ) d ϕ ,
( Δ 2 Φ ) , ϕ | ϕ = ( Δ 2 Φ ( ) ) ϕ | 𝒫 ( | ϕ ) = 0 2 π 𝒫 ( ϕ | ) 𝒫 ( | ϕ ) ( ϕ Φ ( ) ) 2 d ϕ ,
𝒫 ( ϕ | ) 𝒫 ( | ϕ ) = 𝒫 ( ϕ , | ϕ ) .
( Δ 2 Φ ) , ϕ , ϕ = 0 2 π ( Δ 2 Φ ) , ϕ | ϕ 𝒫 ( ϕ ) d ϕ = 0 2 π 0 2 π 𝒫 ( ϕ | ) 𝒫 ( | ϕ ) ( ϕ Φ ( ) ) 2 𝒫 ( ϕ ) d ϕ d ϕ = 0 2 π 𝒫 ( ϕ | ) 𝒫 ( ) ( ϕ Φ ( ) ) 2 d ϕ ,
𝒫 ( ) = 0 2 π 𝒫 ( | ϕ ) 𝒫 ( ϕ ) d ϕ .
( Δ 2 Φ ) , ϕ = 0 2 π 𝒫 ( , ϕ ) ( ϕ Φ ( ) ) 2 d ϕ ,
𝒫 ( , ϕ ) = 𝒫 ( ϕ | ) 𝒫 ( ) .
𝒫 ( | α 0 | ϕ 0 ) = 1 4 ( 1 + e Δ 2 cos ( π / 4 ) ) = 1 4 ( 1 + 2 2 e Δ 2 ) , 𝒫 ( | α 1 | ϕ 1 ) = 1 4 ( 1 + e Δ 2 sin ( 3 π / 4 ) ) = 1 4 ( 1 + 2 2 e Δ 2 ) , 𝒫 ( | α 2 | ϕ 2 ) = 1 4 ( 1 + e Δ 2 cos ( 5 π / 4 ) ) = 1 4 ( 1 + 2 2 e Δ 2 ) , 𝒫 ( | α 3 | ϕ 3 ) = 1 4 ( 1 + e Δ 2 sin ( 7 π / 4 ) ) = 1 4 ( 1 + 2 2 e Δ 2 ) .
𝒫 i i = 𝒫 ( | α i | ϕ i ) ,
j = 0 3 𝒫 ( | α i | ϕ j ) = 1 .
𝒫 i j = j = 0 3 𝒫 ( | α i | ϕ j ) ,
𝒫 i j = 1 𝒫 i i .
QBER = n 𝒬 n n ,
𝒬 n = { k = ( n + 1 ) / 2 n C n k 𝒫 i j k ( 1 𝒫 i j ) n k n is odd , k = ( n + 2 ) / 2 n C n k 𝒫 i j k ( 1 𝒫 i j ) n k + 1 2 C n n / 2 𝒫 i j n / 2 ( 1 𝒫 i j ) n / 2 n is even .
n = e N c N c n n ! , C n k = n ! k ! ( n k ) ! n ! = 1 × 2 × 3 × ( n 1 ) × n ,
Δ ρ = { Δ ρ 1 , Δ ρ 2 , Δ ρ c } .
Δ ρ = ρ s ρ n ,
ρ s = { ρ s 1 , ρ s 2 , , ρ s c } ,
ρ n = { ρ n 1 , ρ n 2 , , ρ n c } ,
PC = { PC 1 , PC 2 , , PC c } .
PC i = 𝒫 ( x s > x n ) = Φ ( Δ ρ i 2 σ 2 ) ,
k i Binomial ( PC i , T ) ,
K = β I ( x : y ) S ( y : E ) ,
I ( x : y ) = 1 2 log 2 V + χ tot 1 + χ tot ,
S ( y : E ) = G ( λ 1 1 2 ) + G ( λ 2 1 2 ) G ( λ 1 3 2 ) G ( λ 4 1 2 ) ,
λ 1 , 2 = 1 2 ( A ± A 2 4 B ,
A = V 2 + T 2 ( V + χ line 2 2 T Z 4 2 ) , B = ( T V 2 + T V χ line T Z 4 2 ) 2 .
λ 3 , 4 = 1 2 ( C ± C 2 4 D ,
C = A χ hom + V B + T ( V + χ line ) T ( V + χ tot ) , D = B V + B χ hom T ( V + χ tot ) .

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