Detector-decoy high-dimensional quantum key distribution

Haize Bao; Wansu Bao; Yang Wang; Ruike Chen; Chun Zhou; Musheng Jiang; Hongwei Li

doi:10.1364/OE.24.022159

1. Introduction

Quantum key distribution (QKD) allows two authorized parties, called Alice and Bob, to share private and secure information at a long distance [1, 2]. Since BB84 protocol [3] was proposed, lots of work on enhancing the security of QKD has been done, such as measurement-device-independent QKD [4–8] and round-robin differential phase-shift QKD [9]. Compared with two-level QKD protocols, high-dimensional quantum key distribution (HD-QKD) [10] enables two parties to generate a secret key at a higher rate. By encoding information in a high-dimensional photonic degrees of freedom, such as position-momentum [10], time-energy [11–15] and orbital angular momentum [16–19], HD-QKD can share more information securely per detected single-photon between two parties so as to improve the secret-key capacity under realistic technical constraints. Moreover, HD-QKD protocols could tolerate more noise than qubit QKD protocols [20].

Recently, based on time-energy entanglement and dispersive optics, a so-called dispersive optic HD-QKD [21] was proposed. It has been proven that the protocol is secure against all collective attacks on the fact that Gaussian attacks are optimum given an experimentally determined covariance matrix [22]. However, like most two-level general QKD protocols, the security of HD-QKD would be influenced by some practical imperfection. Under these imperfect conditions, the imperfect single-photon source always emits multipair events, which causes the HD-QKD vulnerable to the photon number splitting (PNS) attack [23–25] over lossy channels.

To avoid the PNS attack, decoy-state method is designed [26] and developed [27–32]. The central idea of the decoy-state method is to estimate the channel transmission properties by choosing source intensity settings independently and randomly. Recently, Zhenshen Zhang et al. [22] extended decoy-state analysis to HD-QKD protocols with infinite number of decoy states. Darius Bunandar et al. [33] proposed a finite decoy-state HD-QKD protocols and its finite-key analysis was presented in [34].

The detector-decoy method is proposed firstly by Moroder et al. [35] to estimate the photon number distribution, which have showed the advantages and feasibility in QKD. In this paper, by modifying Alice’s detection setups simply, we introduce the detector-decoy method into the HD-QKD protocol. Instead of estimating the photon number distribution, using the detector-decoy method, we propose a new HD-QKD protocol and establish a new security analysis to defend PNS attacks. So our protocol can be considered as an alternative protocol to the decoy-state HD-QKD protocol. Compared with the original decoy-state HD-QKD protocol, without varying the source intensity and optimizing decoy-state intensity, the lower bound of single-photon fraction of postselected events and the upper bound of the leaked information can still be obtained in the detector-decoy HD-QKD. The basic idea of the method is to measure the incoming light pulses with a set of detectors with different efficiencies. We use a variable attenuator to change the transmittance of channel in Alice’s side so that the detector efficiency can be changed. Borrowing realistic experimental parameters, we show by numerical evaluations that in the infinite-size regime the detector-decoy HD-QKD could perform much better than the one-decoy-state HD-QKD. Besides, it performs as well as the two-decoy-state HD-QKD protocol, which can perform as well as the HD-QKD protocol with an infinite number of decoy states. In the finite-size regime, the detector-decoy HD-QKD can obtain much better results. Although our method is considered in the DO-QKD, the same arguments are also applicable to other HD-QKD based on time-energy entanglement.

The paper is organized as follows. In Sec. II, we describe the detector-decoy HD-QKD protocol. In Sec. III, we propose a security analysis for HD-QKD. Then, in Sec. IV we show the result of a numerical evaluation with realistic experimental constraints. Finally, section V. concludes the paper with a summary.

2. Protocol description

In the original decoy-state protocol [33], there is an assumption that Eve cannot intrude into Alice and Bob’s experimental setups. This means that Eve just can attack the quantum channel. The source is located in Alice’s side, so we could consider the photon kept by Alice cannot be affected by Eve. The assumption is also needed in this paper.

Based on detector-decoy method, we propose a detector-decoy HD-QKD scheme by modifying the Alice’s setup slightly as shown in Fig. 1. Specially, our scheme only need one intensity of the source.

Fig. 1 Schematic of the detector-decoy HD-QKD setup. The source is located in Alice’s side. Alice keeps one photon of the pair and sends the other to Bob. They both choose to measure in either the arrival-time basis (case 1) or the frequency basis (case 2) independently and randomly. Their results are only correlated or anti-correlated when they choose the same basis. VA is a variable attenuator. ND is normal dispersion and AD is anomalous dispersion.

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A time-energy entangled state could be produced by Alice weakly pumping the SPDC source. We define σ_cor as the correlation time between two photons, which is determined by the phase matching bandwidth of the SPDC source. Correspondingly, we define σ_coh as the coherence time of the pump field, which is typically far larger than σ_cor. Thus, the number of alphabet characters per photon pulse, d = σ_coh/σ_cor (the Schmidt number) is large [36,37].

The protocol is described as follow:

State preparation: Alice chooses a transmittance of the VA randomly and independently and then generates a biphoton state by SPDC. She keeps one of the biphoton and sends the other to Bob via a quantum channel.
State measurement: Alice selects a basis between time basis and frequency basis randomly and then measures her photon. After receiving the photon from Alice, Bob also performs measurement as Alice and records the outcome.
Classical information post-processing: After all signals are transmitted, Alice publishes her transmittance choice and both of them publish their basis choices over an authenticated public channel. They establish their distilled key only from correlated events acquired in the same basis. Alice and Bob would announce some their measurement results to determine postselection probabilities and excess-noise multipliers. Then Eve’s influence could be detected and they can determine their information advantage over Eve. If the advantage is much greater than zero, they then apply error correction and privacy amplification [36] on their data. As a result, some amount of secret key can be established.

3. Security analysis

In the decoy-state HD-QKD protocol, the postselection probability can be written as [33]

P_{μ} = \sum_{n = 0}^{\infty} \Pr_{n} γ_{n} .

Here, μ is the intensity of Alice’s SPDC source and Pr_n is the probability of generating n–photon pairs. γ_n is the conditional probability of measuring at least one coincident detection event given n–photon pairs are emitted. In our protocol, without Eve’s effect, γ_n can be written explicitly as

γ_{n} = α_{n}^{(η)} β_{n} = [1 - {(1 - η η_{Alice})}^{n} (1 - p_{d})] [1 - {(1 - η_{Bob} η_{T})}^{n} (1 - p_{d})] .

Here,

α_{n}^{(η)} = [1 - {(1 - η η_{Alice})}^{n} (1 - p_{d})]

is the conditional probability of Alice registering at least one detection when the transmittance of the variable attenuator is η and β_n = [1 − (1 −η_Bobη_T)ⁿ (1 − p_d)] is the conditional probability of Bob registering at least one detection in a single measurement frame. η_Alice and η_Bob are Alice and Bob’s detector efficiencies respectively. η_T is is the transmittance of the quantum channel and p_d is the dark count rate in a single measurement frame. Eve, in principle, has the ability to affect the γ_n values by Eve’s effect only on β_n while

α_{n}^{(η)}

is cannot be affected due to the assumption.

The bound on the secure-key capacity is [22,27,33,39]

Δ I \geq β I (A; B) - (1 - F_{μ}^{(η)}) n_{R} - F_{μ}^{(η)} χ_{ζ_{t}, ζ_{ω}}^{U B} (A; E) .

Here,

F_{μ}^{(η)} = \Pr_{1} α_{1}^{(η)} β_{1} / P_{μ}

is the fraction of postselected events that are due to single photon emissions, ζ_t and ζ_ω are the value of time excess noise and frequency excess noise respectively, which are caused by multiphoton emissions, dark counts and Eve’s intrusion. β is reconciliation efficiency and I (A; B) is the mutual information between Alice and Bob. n_R is the number of bits per frame shared by Alice and Bob after error correction is carried out.

χ_{ζ_{t}, ζ_{ω}}^{U B} (A; E)

is an upper bound on Eve’s Holevo information under collective attacks due to the excess noise.

In our scheme, for simplicity, we use photon states under the transmittance η₁ to generate keys. We consider that the statistics of entangled photon-pair produced by SPDC source are approximately Poissonian [40]. So in a single measurement frame, when the mean photon-pair number is μ

\Pr_{n} = \frac{μ^{n}}{n!} e^{- μ} .

To estimate Eve’s effect on β_n, we request Alice to randomly change the transmittance of the variable attenuator η between η₁ and η₂ (0 ≤ η₂ < η₁ ≤ 1).

3.1. Lower bound on $F_{μ}^{(η_{1})}$

We define $P_{μ}^{(η)}$ as the postselection probabilities when the transmittance is η. The postselection probabilities under the two different transmittance η₁ and η₂ (0 ≤ η₂ < η₁ ≤ 1) could be written by

P_{μ}^{(η_{1})} = \sum_{n = 0}^{\infty} \Pr_{n} α_{n}^{(η_{1})} β_{n} = \sum_{n = 0}^{\infty} \frac{μ^{n}}{n!} e^{- μ} (1 - {(1 - η_{1} η_{Alice})}^{n} (1 - p_{d})) β_{n},

P_{μ}^{(η_{2})} = \sum_{n = 0}^{\infty} \Pr_{n} α_{n}^{(η_{2})} β_{n} = \sum_{n = 0}^{\infty} \frac{μ^{n}}{n!} e^{- μ} (1 - {(1 - η_{2} η_{Alice})}^{n} (1 - p_{d})) β_{n} .

Specially, when η = 1,

P_{μ}^{(η)}

is equivalent to P_μ in essence.

Then we will use the Eq. (5) and Eq. (6) to deduce a lower bound of β₁.

\begin{matrix} \frac{e^{μ} P_{μ}^{(η_{1})}}{1 - {(1 - η_{1} η_{Alice})}^{2} (1 - p_{d})} - \frac{e^{μ} P_{μ}^{(η_{2})}}{1 - {(1 - η_{2} η_{Alice})}^{2} (1 - p_{d})} \\ = β_{0} [\frac{p_{d}}{1 - {(1 - η_{1} η_{Alice})}^{2} (1 - p_{d})} \frac{p_{d}}{1 - {(1 - η_{2} η_{Alice})}^{2} (1 - p_{d})}] \\ + β_{1} [\frac{1 - (1 - η_{1} η_{Alice}) (1 - p_{d})}{1 - {(1 - η_{1} η_{Alice})}^{2} (1 - p_{d})} μ - \frac{1 - (1 - η_{2} η_{Alice}) (1 - p_{d})}{1 - {(1 - η_{2} η_{Alice})}^{2} (1 - p_{d})} μ] \\ + \sum_{n = 3}^{\infty} β_{n} [\frac{1 - {(1 - η_{1} η_{Alice})}^{n} (1 - p_{d})}{1 - {(1 - η_{1} η_{Alice})}^{2} (1 - p_{d})} μ^{n} - \frac{1 - {(1 - η_{2} η_{Alice})}^{n} (1 - p_{d})}{1 - {(1 - η_{2} η_{Alice})}^{2} (1 - p_{d})} μ^{n}] . \end{matrix}

The inequality

\frac{1 - {(1 - η_{1} η_{Alice})}^{n} (1 - p_{d})}{1 - {(1 - η_{2} η_{Alice})}^{n} (1 - p_{d})} \leq \frac{1 - {(1 - η_{1} η_{Alice})}^{2} (1 - p_{d})}{1 - {(1 - η_{2} η_{Alice})}^{2} (1 - p_{d})}

for n ≥ 2 is satisfied given 0.018 ≤ η₂η_Alice < η₁η_Alice < 1, which is easily to be realized. That means the equation

\sum_{n = 3}^{\infty} β_{n} [\frac{1 - {(1 - η_{1} η_{Alice})}^{n} (1 - p_{d})}{1 - {(1 - η_{1} η_{Alice})}^{2} (1 - p_{d})} μ^{n} - \frac{1 - {(1 - η_{2} η_{Alice})}^{n} (1 - p_{d})}{1 - {(1 - η_{2} η_{Alice})}^{2} (1 - p_{d})} μ^{n}] \leq 0 .

Then Eq. (9) leads to the following inequality:

β_{1} \geq β_{1}^{L B} = \frac{\frac{e^{μ} P_{μ}^{(η_{1})}}{1 - {(1 - η_{1} η_{Alice})}^{2} (1 - p_{d})} - \frac{e^{μ} P_{μ}^{(η_{2})}}{1 - {(1 - η_{2} η_{Alice})}^{2} (1 - p_{d})} + β_{0} [\frac{p_{d}}{1 - {(1 - η_{1} η_{Alice})}^{2} (1 - p_{d})} - \frac{p_{d}}{1 - {(1 - η_{2} η_{Alice})}^{2} (1 - p_{d})}]}{\frac{1 - (1 - η_{1} η_{Alice}) (1 - p_{d})}{1 - {(1 - η_{1} η_{Alice})}^{2} (1 - p_{d})} - \frac{1 - (1 - η_{2} η_{Alice}) (1 - p_{d})}{1 - {(1 - η_{2} η_{Alice})}^{2} (1 - p_{d})}} .

Now we need to lower bound β₀. As it’s shown in [22, 33], under the assumption that Eve cannot intrude into both experimental setups, the probability β₀ cannot be lower than the dark count rate no matter what Eve does:

β_{0} \geq β_{0}^{L B} = p_{d} .

Therefore, we can give rise to the fraction of postselected events that are due to single photon emissions as:

\begin{matrix} F_{μ}^{(η_{1})} = α_{1}^{(η_{1})} β_{1} \frac{μ e^{- μ}}{P_{μ}^{(η_{1})}} \geq F_{μ}^{L B} = (1 - (1 - η_{1} η_{Alice}) (1 - p_{d})) \\ \times \frac{\frac{P_{μ}^{(η_{1})}}{1 - {(1 - η_{1} η_{Alice})}^{2} (1 - p_{d})} - \frac{P_{μ}^{(η_{2})}}{1 - {(1 - η_{2} η_{Alice})}^{2} (1 - p_{d})} + e^{- μ} β_{0}^{L B} [\frac{p_{d}}{1 - {(1 - η_{1} η_{Alice})}^{2} (1 - p_{d})} - \frac{p_{d}}{1 - {(1 - η_{2} η_{Alice})}^{2} (1 - p_{d})}]}{P_{μ}^{(η_{1})} [\frac{1 - (1 - η_{1} η_{Alice}) (1 - p_{d})}{1 - {(1 - η_{1} η_{Alice})}^{2} (1 - p_{d})} - \frac{1 - (1 - η_{2} η_{Alice}) (1 - p_{d})}{1 - {(1 - η_{2} η_{Alice})}^{2} (1 - p_{d})}]} \end{matrix}

3.2. Lower bound on ζ_t and ζ_ω

When Eve attacks Alice’s transmission, the decrease in measurement correlations of Alice and Bob is caused, which is parameterized by the excess-noise factors ζ_t and ζ_ω.

We consider the relation in [33]. Ω_x (for x = t and ω) is the averaged excess-noise multiplier, which can be measured by two authenticated parties and used to estimate ζ_t and ζ_ω by following equations:

Ω_{x} = F_{μ}^{(η)} (1 + ζ_{x}) + Δ Ω_{x} (1 - F_{μ}^{(η)}),

where x = t and ω.

Equation (13) can be divided into two groups by transmittance η₁ and η₂, which can be expressed as:

Ω_{x, η_{1}} = F_{μ}^{(η_{1})} (1 + ζ_{x}) + Δ Ω_{x} (1 - F_{μ}^{(η_{1})}),

Ω_{x, η_{2}} = F_{μ}^{(η_{2})} (1 + ζ_{x}) + Δ Ω_{x} (1 - F_{μ}^{(η_{2})}) .

We multiply these two equations by

P_{μ}^{(η_{1})} e^{μ}

and

P_{μ}^{(η_{2})} e^{μ}

respectively, and then we have

Ω_{x, η_{1}} P_{μ}^{(η_{1})} e^{μ} = μ α_{1}^{(η_{1})} β_{1} (1 + ζ_{x}) + Δ Ω_{x} (P_{μ}^{(η_{1})} e^{μ} - μ α_{1}^{(η_{1})} β_{1}),

Ω_{x, η_{2}} P_{μ}^{(η 2)} e^{μ} = μ α_{1}^{(η 2)} β_{1} (1 + ζ_{x}) + Δ Ω_{x} (P_{μ}^{(η 2)} e^{μ} - μ α_{1}^{(η_{2})} β_{1}) .

Combing Eq. (16) and Eq. (17), we get

\begin{matrix} Ω_{x, η_{1}} P_{μ}^{(η_{1})} e^{μ} - Ω_{x, η_{2}} P_{μ}^{(η_{2})} e^{μ} \\ = (α_{1}^{(η_{1})} - α_{1}^{(η_{2})}) μ β_{1} (1 + ζ_{x}) + Δ Ω_{x} [P_{μ}^{(η_{1})} e^{u} - P_{μ}^{(η_{2})} e^{u} - (α_{1}^{(η_{1})} - α_{1}^{(η_{2})}) μ β_{1}] \\ = (η_{1} - η_{2}) η_{Alice} (1 - p_{d}) μ β_{1} (1 + ζ_{x}) + Δ Ω_{x} [P_{μ}^{(η_{1})} e^{u} - P_{μ}^{(η_{2})} e^{u} - (η_{1} - η_{2}) η_{Alice} (1 - p_{d}) μ β_{1}] \\ \geq (η_{1} - η_{2}) η_{Alice} (1 - p_{d}) μ β_{1} (1 + ζ_{x}), \end{matrix}

where the inequality comes from

\begin{matrix} P_{μ}^{(η_{1})} - P_{μ}^{(η_{2})} = \sum_{n = 0}^{\infty} {[{(1 - η_{2} η_{Alice})}^{n} - {(1 - η_{1} η_{Alice})}^{n}] (1 - p_{d})} β_{n} \\ \geq (η_{1} - η_{2}) η_{Alice} (1 - p_{d}) μ β_{1} . \end{matrix}

Thus,

(1 + ζ_{x}) \leq \frac{(Ω_{x, η_{1}} P_{μ}^{(η_{1})} - Ω_{x, η_{2}} P_{μ}^{(η_{2})}) e^{μ}}{(η_{1} - η_{2}) η_{Alice} (1 - p_{d}) μ β_{1}} \leq \frac{(Ω_{x, η_{1}} P_{μ}^{(η_{1})} - Ω_{x, η_{2}} P_{μ}^{(η_{2})}) e^{μ}}{(η_{1} - η_{2}) η_{Alice} (1 - p_{d}) μ β_{1}^{L B}}

for x = t and ω.

The upper bounds on ζ_t and ζ_ω could be obtained from:

\begin{matrix} Ω_{x, η} & = F_{μ}^{(η)} (1 + ζ_{x}) + Δ Ω_{x} (1 - F_{μ}^{(η)}) \geq F_{μ}^{(η)} (1 + ζ_{x}) \\ = \frac{[1 - (1 - η η_{Alice}) (1 - p_{d})] P_{μ}^{(η_{1})}}{[1 - (1 - η_{1} η_{Alice}) (1 - p_{d})] P_{μ}^{(η)}} F_{μ}^{(η_{1})} (1 + ζ_{x}) \\ \geq \frac{[1 - (1 - η η_{Alice}) (1 - p_{d})] P_{μ}^{(η_{1})}}{[1 - (1 - η η_{Alice}) (1 - p_{d})] P_{μ}^{(η)}} F_{μ}^{L B} (1 + ζ_{x}) \end{matrix}

for η = η₁ or η₂. This inequality implies that

(1 + ζ_{x}) \leq \frac{[1 - (1 - η η_{Alice}) (1 - p_{d})] P_{μ}^{(η)} Ω_{x, η}}{[1 - (1 - η η_{Alice}) (1 - p_{d})] P_{μ}^{(η_{1})} F_{μ}^{L B}} .

Therefore, we can give the upper bounds on ζ_t and ζ_ω as follows:

\begin{array}{l} ζ_{x} \leq ζ_{x}^{U B} = min {\frac{(Ω_{x, η_{1}} P_{μ}^{(η_{1})} - Ω_{x, η_{2}} P_{μ}^{(η_{2})}) e^{μ}}{(η_{1} - η_{2}) η_{Alice} (1 - p_{d}) μ β_{1}^{L B}}, \\ min_{η \in {η_{1}, η_{2}}} {\frac{[1 - (1 - η_{1} η_{Alice}) (1 - p_{d})] P_{μ}^{(η)} Ω_{x, η}}{[1 - (1 - η η_{Alice}) (1 - p_{d})] P_{μ}^{(η_{1})} F_{μ}^{L B}}}} - 1 . \end{array}

Now with Eq. (12) and Eq. (23), we obtain a lower bound on F_μ and upper bound on ζ_t and ζ_ω.

4. Numerical evaluation

We borrow some realistic experimental parameters in the implementation of the HD-QKD [41]: propagation loss α = 0.2 dB/km, detector timing jitter σ_J = 20 ps, dark count rate R_dc = 1000 s⁻¹, reconciliation efficiency β = 0.9, n_R = log₂ d. Therefore, we could know the channel transmittance by the function η_T = 10^−αL/10. The detector efficiencies of Alice and Bob’s setups are assumed to be equal. Here we assume that the detection efficiencies are η_Alice = η_Bob = 0.93 [42].

We take coherent time σ_cor = 30 ps for all d values and correspondingly σ_coh = dσ_cor, which is easily controlled. We choose a single measurement frame duration $T_{f} = 2 \sqrt{ln 2} σ_{coh}$ [22]. For simplicity, we consider two excess-noise factors are equal, ζ_t = ζ_ω = ζ. The change in correlation time due to Eve’s interaction can be assumed to be $Δ σ = (\sqrt{1 + ζ} - 1)$ σ_cor = 10 ps [43], which doesn’t lose generality.

In our detector-decoy method, we vary the transmittance of the variable attenuator between η₁ and η₂, specifically η₁ = 1 and η₂ = 0.5. Here, we assume that events in which both Alice and Bob click when η₁ = 1 are used to encode information.

Here, in the infinite-size regime we give a comparison between the detector-decoy HD-QKD protocol and the decoy-state HD-QKD protocol with average photon number μ = 0.1 when the dimension d = 8 and d = 32, which is shown in Fig. 2 As one can see, the detector-decoy method can perform much better than one-decoy-state HD-QKD and as well as the two-decoy-state HD-QKD protocol, which means that the detector-decoy HD-QKD could work as the decoy-state HD-QKD with infinite number of decoy states without the source intensity modulation or monitoring photon numbers [34].

Fig. 2 Lower bounds on the secure-key capacity in bits per coincidence for (a) d = 8 and (b) d = 32 as a function of transmission distance at 10 km increments in the infinte-size regime. The results show that the detector-decoy method (black dash lines) could perform much better than one-decoy-state HD-QKD (blue dash lines) and as well as the two-decoy-state HD-QKD (red dotted lines).

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Besides, we also give a comparison between the detector-decoy HD-QKD protocol and the two-decoy-state HD-QKD protocol in the finite-size regime, which is more practical in realistic constraints. In the finite-size regime, the postselection probability $P_{μ}^{(η)}$ should satisfy the relationship:

{P_{μ}^{(η)}}^{\pm} = P_{μ}^{(η)} \pm Δ .

Here, Δ is the fluctuation range of the postselection, which is related the number of frames. Using Eq. (24), and following the approach proposed in [34], we can show the performance of the detector-decoy HD-QKD protocol and the two-decoy-state HD-QKD protocol in the finite-size regime. As is shown in Fig. 3, the dector-decoy HD-QKD protocol can achieve much better effects than the decoy-state HD-QKD protocol.

Fig. 3 Lower bounds on the secure-key capacity in the finite-size regime for pulse number N = 10¹² and d = 8. The results show that the detector-decoy method (black dash lines) could perform much better than two-decoy-state HD-QKD (red dash lines).

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In general, the efficiency of conventional detectors used in QKD experiment is always far lower than the efficiency of the superconducting nanowire single-photon detector [44]. So, we also give a comparison between schemes in the condition that is limited with low detector efficiencies, especially η_Alice = η_Bob = 0.045 shown in Fig. 4. Figure 4(a) plots the lower bound values on F_μ that are obtained by detector-decoy method (blue solid lines) and two-decoy-state HD-QKD (red solid lines) respectively. Figure 4(b) shows the comparison between two protocols. Both of them are plotted under the condition with d = 8 and μ = 0.10. As the figure shown, when the detector efficiency is low, the advantage of detector-decoy HD-QKD become prominent. That means the detector-decoy HD-QKD could tolerate lower detector efficiency.

Fig. 4 (a) The lower bound values on F_μ obtained by detector-decoy method (blue solid lines) and two-decoy-state HD-QKD (red solid lines). (b) The secret key capacity per coincidence for original decoy-state HD-QKD (red dotted lines) and detector-decoy HD-QKD (black solid lines) as a function of the transmission distance when d = 8, μ = 0.10 and η_Alice = η_Bob = 0.045.

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5. Conclusion

The decoy-state method is applied to resist PNS attacks in HD-QKD systems. The original decoy-state HD-QKD has proved that the two-decoy-state protocol can perform as well as the protocol with infinite decoy states. In this paper, we proposed a detector-decoy HD-QKD protocol that could offer certain practical advantages over the original decoy-state HD-QKD protocol. In the infinite-size regime, our scheme, using a variable attenuator, could achieve secure information per coincidence as the two-decoy-state HD-QKD with little or no compromise. In our protocol, operations of choosing and optimizing the source intensity of decoy states are omitted. This is a main advantage over the original protocol. Besides, in finite-size regime, our protocol is much more practical than the original protocol. Specially, if the detector efficiency is lower, the advantage of our method become more prominent.

In realistic constraints, the detector-decoy HD-QKD presented in this paper provides a more applicable and feasible scheme in the practical implementation of the HD-QKD protocol with slight modification. The source intensity in our scheme is constant and correspondingly we do not need to optimize the intensity of decoy states. When the HD-QKD protocol is implemented practically, our scheme can perform much better than the original two-decoy-state HD-QKD protocol but operations are much simpler.

In practical implementation, the performance of QKD systems can be influenced by many constraints. So it’s necessary to conduct a further study on how to optimize the detector decoy method in the HD-QKD protocol under different constraints. What’s more, besides the DO-QKD protocol, many other HD-QKD protocols have been proposed. Our future work will focus on extending the detector-decoy method to other HD-QKD protocols and studying if the performance advantage of detector-decoy method is universal over all decoy-state HDQKD protocols in the finite-size regime.

Funding

National Basic Research Program of China (2013CB338002); National Natural Science Foundation (NSFC) (11304397 and 61505261).

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Detector-decoy high-dimensional quantum key distribution

Abstract

1. Introduction

2. Protocol description

3. Security analysis

3.1. Lower bound on $F_{μ}^{(η_{1})}$

3.2. Lower bound on ζ_t and ζ_ω

4. Numerical evaluation

5. Conclusion

Funding

References and links

Cited By

Figures (4)

Equations (24)

Optics Express

Abstract

1. Introduction

2. Protocol description

3. Security analysis

3.1. Lower bound on Fμ(η1)

3.2. Lower bound on ζt and ζω

4. Numerical evaluation

5. Conclusion

Funding

References and links

Cited By

Figures (4)

Equations (24)

Optics Express

3.1. Lower bound on $F_{μ}^{(η_{1})}$

3.2. Lower bound on ζ_t and ζ_ω