Phase noise estimation using Bayesian inference for continuous-variable quantum key distribution

Wei Zhao; Ying Guo; Ling Zhang; Duan Huang

doi:10.1364/OE.27.001838

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 |α_k〉 = |αe^i(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 |α_k〉 = |αe^i(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 |α_i+1〉 is the phase shift of π/2 relative to the state |α_i〉.

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 |α_i〉 are denoted as n_i 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 $N_{c} = \sum_{i = 0}^{3} n_{i}$ . 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 ℰ ∈ {|α₀〉, |α₁〉, |α₂〉, |α₃〉} is obtained by PNRD, so that the posterior probability distribution takes the following form [26]

𝒫 (ϕ | ℰ) = \frac{𝒫 (ℰ | ϕ) 𝒫 (ϕ)}{\int 𝒫 (ℰ | ϕ) 𝒫 (ϕ) d ϕ} .

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]

\begin{array}{l} 𝒫 (| α_{0} 〉 | ϕ) = \frac{1}{4} (1 + e^{- Δ^{2}} cos (ϕ)), \\ 𝒫 (| α_{1} 〉 | ϕ) = \frac{1}{4} (1 + e^{- Δ^{2}} sin (ϕ)), \\ 𝒫 (| α_{2} 〉 | ϕ) = \frac{1}{4} (1 - e^{- Δ^{2}} cos (ϕ)), \\ 𝒫 (| α_{3} 〉 | ϕ) = \frac{1}{4} (1 - e^{- Δ^{2}} sin (ϕ)) . \end{array}

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]

𝒫_{B} (ϕ; 𝒩) = \frac{\sum_{i = 0}^{3} 𝒫 {(| α_{i} 〉 | ϕ)}^{n_{i}}}{ℳ},

where ℳ is the normalization factor satisfies that

\int_{0}^{2 π} 𝒫_{B} (ϕ; 𝒩) d ϕ = 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 n_i = 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 ℰ 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(ℰ) of ϕ′ can be made as [32]

Φ (ℰ) = \int_{0}^{2 π} ϕ 𝒫 (ϕ | ℰ) d ϕ .

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

{(Δ^{2} Φ (ℰ))}_{ϕ | ℰ} = \int_{0}^{2 π} {(ϕ - Φ (ℰ))}^{2} 𝒫 (ϕ | ℰ) d ϕ,

which is considered as the measurement of fluctuation about Φ(ℰ). Nevertheless, the outcome ℰ in Eq. (6) is an arbitrary measurement sequence. In other words, the definition of variance relies on the specific ℰ. Hence, averaging over the measurement sequences is necessary. Consequently, the average Bayesian posterior variance can be written as

\begin{array}{l} {(Δ^{2} Φ)}_{ℰ, ϕ | ϕ^{'}} = \sum_{ℰ} {(Δ^{2} Φ (ℰ))}_{ϕ | ℰ} 𝒫 (ℰ | ϕ^{'}) \\ \begin{array}{l} = \sum_{ℰ} \int_{0}^{2 π} 𝒫 (ϕ | ℰ) 𝒫 (ℰ | ϕ^{'}) {(ϕ - Φ (ℰ))}^{2} d ϕ, \end{array} \end{array}

where

𝒫 (ϕ | ℰ) 𝒫 (ℰ | ϕ^{'}) = 𝒫 (ϕ, ℰ | ϕ^{'}) .

Afterwards, we average over 𝒫(ϕ′) of the Bayesian posterior variance in Eq. (7), the expression can be described as

\begin{array}{l} {(Δ^{2} Φ)}_{ℰ, ϕ, ϕ^{'}} = \int_{0}^{2 π} {(Δ^{2} Φ)}_{ℰ, ϕ | ϕ^{'}} 𝒫 (ϕ^{'}) d ϕ^{'} \\ \begin{array}{l} = \sum_{ℰ} \int_{0}^{2 π} \int_{0}^{2 π} 𝒫 (ϕ | ℰ) 𝒫 (ℰ | ϕ^{'}) {(ϕ - Φ (ℰ))}^{2} 𝒫 (ϕ^{'}) d ϕ^{'} d ϕ \end{array} \\ \begin{array}{l} = \sum_{ℰ} \int_{0}^{2 π} 𝒫 (ϕ | ℰ) 𝒫 (ℰ) {(ϕ - Φ (ℰ))}^{2} d ϕ, \end{array} \end{array}

with the notation

𝒫 (ℰ) = \int_{0}^{2 π} 𝒫 (ℰ | ϕ^{'}) 𝒫 (ϕ^{'}) d ϕ^{'} .

Considering the fluctuation of ϕ′, the probability 𝒫(ℰ) is the average probability to observe the measurement sequence ℰ.

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

{(Δ^{2} Φ)}_{ℰ, ϕ} = \sum_{ℰ} \int_{0}^{2 π} 𝒫 (ℰ, ϕ) {(ϕ - Φ (ℰ))}^{2} d ϕ,

where

𝒫 (ℰ, ϕ) = 𝒫 (ϕ | ℰ) 𝒫 (ℰ) .

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

\begin{array}{l} 𝒫 (| α_{0} 〉 | ϕ_{0}) = \frac{1}{4} (1 + e^{- Δ^{2}} cos (π / 4)) = \frac{1}{4} (1 + \frac{\sqrt{2}}{2} e^{- Δ^{2}}), \\ 𝒫 (| α_{1} 〉 | ϕ_{1}) = \frac{1}{4} (1 + e^{- Δ^{2}} sin (3 π / 4)) = \frac{1}{4} (1 + \frac{\sqrt{2}}{2} e^{- Δ^{2}}), \\ 𝒫 (| α_{2} 〉 | ϕ_{2}) = \frac{1}{4} (1 + e^{- Δ^{2}} cos (5 π / 4)) = \frac{1}{4} (1 + \frac{\sqrt{2}}{2} e^{- Δ^{2}}), \\ 𝒫 (| α_{3} 〉 | ϕ_{3}) = \frac{1}{4} (1 + e^{- Δ^{2}} sin (7 π / 4)) = \frac{1}{4} (1 + \frac{\sqrt{2}}{2} e^{- Δ^{2}}) . \end{array}

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]

𝒫_{i i} = 𝒫 (| α_{i} 〉 | ϕ_{i}),

with i ∈ {0, 1, 2, 3}, which satisfies the constraint

\sum_{j = 0}^{3} 𝒫 (| α_{i} 〉 | ϕ_{j}) = 1 .

According to the Eq. (14) and (15), the transition probability can be expressed as

𝒫_{i j} = \sum_{j = 0}^{3} 𝒫 (| α_{i} 〉 | ϕ_{j}),

with i ≠ j, further simplification can be express as

𝒫_{i j} = 1 - 𝒫_{i i} .

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

QBER = \sum_{n} 𝒬_{n} ℛ_{n},

with the notation

𝒬_{n} = {\begin{array}{l} \sum_{k = (n + 1) / 2}^{n} C_{n}^{k} 𝒫_{i j}^{k} {(1 - 𝒫_{i j})}^{n - k} & n is odd, \\ \sum_{k = (n + 2) / 2}^{n} C_{n}^{k} 𝒫_{i j}^{k} {(1 - 𝒫_{i j})}^{n - k} + \frac{1}{2} C_{n}^{n / 2} 𝒫_{i j}^{n / 2} {(1 - 𝒫_{i j})}^{n / 2} & n is even . \end{array}

and

ℛ_{n} = \frac{e^{- N_{c}} \cdot N_{c}^{n}}{n!}, C_{n}^{k} = \frac{n!}{k! (n - k)!} n! = 1 \times 2 \times 3 \times \dots (n - 1) \times n,

where ℛ_n is the Poisson distribution because that the number of photons satisfies the poisson law [36–38], 𝒫_ij has been defined in Eq. (16),

C_{n}^{k}

and n ! represent the combination and factorial, respectively.

In the Eq. (19), n represents the transmitted photons per pulse and N_c represents the detected photons per pulse. If the channel is lossless, two parameters are satisfied n = N_c. If the channel is loss, two parameters are satisfied n > N_c. Comprehensively considering both the lossless channel and lossy channel, two parameters are subject to the restriction n ≥ N_c. As shown in Fig. 5, the QBER is the function of transmitted photons n and detected photons N_c per pulse. Therefore, Fig. 5 is separated into feasible region (n ≥ N_c) and infeasible region (n < N_c). 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 N_c 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 N_c. If the channel is lossless, two parameters are satisfied n = N_c. If the channel is loss, two parameters are satisfied n > N_c. Comprehensively considering both the lossless channel and lossy channel, two parameters are subject to the restriction n ≥ N_c.

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Fig. 6 The mean quantum bit error rate (QBER). Here, N_c 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.

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

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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

Δ ρ = {Δ ρ_{1}, Δ ρ_{2}, \dots Δ ρ_{c}} .

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] $Δ ρ = ρ_{s} - ρ_{n},$ with $ρ_{s} = {ρ_{s_{1}}, ρ_{s_{2}}, \dots, ρ_{s_{c}}},$ and $ρ_{n} = {ρ_{n_{1}}, ρ_{n_{2}}, \dots, ρ_{n_{c}}},$ For each trial t, one signal item and one noise item will be displayed, with true feature values of ρ_{s_i} and ρ_{n_i}, where ρ_s and ρ_n 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 x_s and x_n, respectively. Subsequently, we define the concept of correct response $PC = {{PC}_{1}, {PC}_{2}, \dots, {PC}_{c}} .$ On each trial t, the correct response are conducted when the signal and phase noise locations satisfies the constraint x_s > x_n. Hence, the correct response can be expressed as [41] ${PC}_{i} = 𝒫 (x_{s} > x_{n}) = Φ (\frac{Δ ρ_{i}}{2 σ^{2}}),$ with i = 1, 2, · · ·, c, where Δρ_i 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, · · · , k_c} 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 k_i is subject to Binomially distributed [44–46], so the parameter satisfy $k_{i} \sim Binomial ({PC}_{i}, T),$ 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, · · ·, k_c/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 χ_line = 1/T′ + ∊ − 1, where T′ and ∊ represent the transmission efficiency and excess noise, respectively. The transmittance is defined as T′ = 10^−αL/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 v_el and efficiency η. The detection-added noise is defined as χ_hom = [(1 − η) + v_el]/η. There, the total noise referred to the channel input can be expressed as χ_tot = χ_line + χ_hom/T.

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

K = β I (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) = \frac{1}{2} {log}_{2} \frac{V + χ_{tot}}{1 + χ_{tot}},

with the notation V = V_A + 1, where V_A is the Alice’s modulation variance. The Holevo bound of the information between Eve and Bob is given by [25]

S (y : E) = G (\frac{λ_{1} - 1}{2}) + G (\frac{λ_{2} - 1}{2}) - G (\frac{λ_{1} - 3}{2}) - G (\frac{λ_{4} - 1}{2}),

where G(x) = (x + 1)log₂(x + 1) − x log₂x. The symplectic eigenvalues λ_1,2 are given by

λ_{1, 2} = \sqrt{\frac{1}{2} (A \pm \sqrt{A^{2} - 4 B},}

where

\begin{array}{l} 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} . \end{array}

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

Z_{G} = \sqrt{V_{A}^{2} + 2 V_{A}}

. The symplectic eigenvalues λ_3,4 are given by [24,25]

λ_{3, 4} = \sqrt{\frac{1}{2} (C \pm \sqrt{C^{2} - 4 D},}

with the notation

\begin{array}{l} C = \frac{A χ_{hom} + V \sqrt{B} + T (V + χ_{line})}{T (V + χ_{tot})}, \\ D = \sqrt{B} \frac{V + \sqrt{B} χ_{hom}}{T (V + χ_{tot})} . \end{array}

Subsequently, we conduct numerical simulation using the realistic parameters. The parameters are summarized below: α = 0.2 dB/km, v_el = 0.1, ∊ = 0.05, η = 0.5, β = 0.95 and V_A = 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, v_el = 0.1, ∊ = 0.05, η = 0.5, β = 0.95 and V_A = 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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The Bayesian phase estimation algorithm	The Bayesian phase estimation theory
Step 1. Draw n samples from 𝒩(μ, σ), namely ϕ_j ∈𝒩(μ, σ);	Step 1. An initial prior distribution is provided to express the confidence of the correct eigenphase in the current hypotheses;
Step 2. Perform experiment and observe outcome ℰ∈ {\|α₀〉, \|α₁〉, \|α₂〉, \|α₃〉};	Step 2. The result of each new measurement ℰ is used to update the mean μ and standard deviation σ, then the updated prior distribution should be $𝒫 (ϕ \| ℰ) = \frac{𝒫 (ℰ \| ϕ) 𝒫 (ϕ)}{\int 𝒫 (ℰ \| ϕ) 𝒫 (ϕ) d ϕ}$ ;
Step 3. For each sample ϕ_j, assign ϕ_j to Φ _accept with probability 𝒫(ℰ\|ϕ_j)/𝒦_E where 𝒦_E ∈ (0, 1] is a constant s.t.𝒫 (ℰ\|ϕ_j)/𝒦_E ≤ 1 for all ϕ_j and ℰ;	Step 3. A host of particles are drawn from the prior distribution and then probabilistically discarded based on the likelihood function;
Step 4. Return phase estimation and its uncertainty.	Step 4. The remaining samples model the posterior distribution. After using Eq. (1) to update the posterior distribution, we then set the prior distribution to equal the posterior distribution. This process is then repeated for each of the random experiments in the dataset.

Input: Initial prior probability distribution 𝒩(μ₀, σ₀), constant 𝒦_E.
Output: the phase estimation mean μ and phase standard derivation σ.
μ = μ₀, σ = σ₀.	% Initialization data.
for i ∈ 1 → m do
Obtain the measurement outcome ℰ from the experiment.
(μ, σ) = Update(ℰ, μ, σ, 𝒦_E).	% Update the prior distribution based on the likelihood function.
end for
return (μ, σ).

Input: phase mean μ and standard derivation σ, measurement outcome ℰ, constant 𝒦_E.
Output: phase estimation mean μ′ and standard derivation σ′ of the posterior distribution.
function Update(ℰ, μ, σ, 𝒦_E)
μ_acc, μ′_acc, V_acc, V′_acc = 0.
for i ∈ 1 → n do
x ∼ 𝒩(μ, σ)
x = x mod 2π.
x′ = x + π mod 2π.
u ∼ Uniform (0, 1).	% The probability 𝒫(ℰ \|x) of the acceptable particle.
if 𝒫(ℰ\|x) ≥ 𝒦_Eu then
μ_acc = μ_acc + x.
V_acc = V_acc + x².
V′_acc = V′_acc + x′².
N_acc = N_acc + 1.
end if
end for
μ′ = μ_acc/N_acc.
$σ^{'} = min (\sqrt{V_{acc} - μ_{acc}^{2}) / (N_{acc} - 1)}$ , $\sqrt{(V_{acc}^{'} - μ_{acc}^{2}) / (N_{acc} - 1))}$ .
return (μ′, σ′).
end function

The Bayesian phase estimation algorithm	The Bayesian phase estimation theory
Step 1. Draw n samples from 𝒩(μ, σ), namely ϕ_j ∈𝒩(μ, σ);	Step 1. An initial prior distribution is provided to express the confidence of the correct eigenphase in the current hypotheses;
Step 2. Perform experiment and observe outcome ℰ∈ {\|α₀〉, \|α₁〉, \|α₂〉, \|α₃〉};	Step 2. The result of each new measurement ℰ is used to update the mean μ and standard deviation σ, then the updated prior distribution should be $𝒫 (ϕ \| ℰ) = \frac{𝒫 (ℰ \| ϕ) 𝒫 (ϕ)}{\int 𝒫 (ℰ \| ϕ) 𝒫 (ϕ) d ϕ}$ ;
Step 3. For each sample ϕ_j, assign ϕ_j to Φ _accept with probability 𝒫(ℰ\|ϕ_j)/𝒦_E where 𝒦_E ∈ (0, 1] is a constant s.t.𝒫 (ℰ\|ϕ_j)/𝒦_E ≤ 1 for all ϕ_j and ℰ;	Step 3. A host of particles are drawn from the prior distribution and then probabilistically discarded based on the likelihood function;
Step 4. Return phase estimation and its uncertainty.	Step 4. The remaining samples model the posterior distribution. After using Eq. (1) to update the posterior distribution, we then set the prior distribution to equal the posterior distribution. This process is then repeated for each of the random experiments in the dataset.

Input: Initial prior probability distribution 𝒩(μ₀, σ₀), constant 𝒦_E.
Output: the phase estimation mean μ and phase standard derivation σ.
μ = μ₀, σ = σ₀.	% Initialization data.
for i ∈ 1 → m do
Obtain the measurement outcome ℰ from the experiment.
(μ, σ) = Update(ℰ, μ, σ, 𝒦_E).	% Update the prior distribution based on the likelihood function.
end for
return (μ, σ).

Phase noise estimation using Bayesian inference for continuous-variable quantum key distribution

Abstract

1. introduction

2. Bayesian quantum inference algorithm

2.1. Four-state CVQKD protocol

2.2. Probability density function of phase shift

2.3. Bayesian posterior variance

3. Quantum bit error rate

4. Model prediction

4.1. Inferences made with phase variance

4.2. Model predicted performance

5. Conclusion

Appendix

Funding

References

Cited By

Figures (10)

Tables (3)

Equations (34)

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