Improvement of self-referenced continuous-variable quantum key distribution with quantum photon catalysis

Wei Ye; Hai Zhong; Qin Liao; Duan Huang; Liyun Hu; Ying Guo

doi:10.1364/OE.27.017186

1. Introduction

Quantum key distribution (QKD) [1–4] provides a secure approach for two distant legal users, Alice and Bob, to share a common secret key over an untrusted channel since its unconditional security has been guaranteed by the fundamental laws of quantum mechanics [5–8]. Continuous-variable (CV) QKD protocol [9], as a promising alternative, encodes information on the quadratures of the optical field and then decodes the secret information by the better efficiency of coherent detection so that this protocol promises higher key rates than its discrete-variable (DV) QKD counterpart [10–15]. Moreover, other advantages of CVQKD including the low experimental cost and the compatibility with current optical fiber communication have reflected its potential application in commerce, especially for the Gaussian modulated coherent state (GMCS) CVQKD protocol initially proposed theoretically in [16] and experimentally in [17].

In the traditional implementations of GMCS-CVQKD protocol [18–20], the quantum signal is co-transmitted with a bright local oscillator (LO) pulse from Alice to Bob through the quantum channel. However, there exist some security holes of this transmitted LO (TLO) design when implementing in practice. For instance, to realize the long distance and the high speed communication, a stronger LO is usually required at the sender’s side, but unfortunately brings about the inefficiency of QKD and the certain security loopholes, such as the LO fluctuation attack [21] and the calibration attack [22]. In order to overcome these problems, a novel self-referenced (SR) CVQKD protocol [23–29] is the most remarkable in which the extra generation of the local LO is placed at receiver’s side for coherent detections. In SR-CVQKD, Alice consecutively sends two optical pulses used as the quantum signal pulse and the phase reference pulse, respectively, and meanwhile at reception, Bob uses the locally generated LO for coherent detections to estimate the relative phase by employing the phase reference. Note that the maximal transmission distance for the traditional SR-CVQKD is associated with the amplitude of the reference pulse. However, in practice, the amplitude of the reference pulse is unable to take infinity, so it is crucial that how to expand the maximal transmission distance especially when the reference pulse is weak.

Fortunately, due to the fact that the entanglement of Gaussian entangled states can be improved by exploiting non-Gaussian operations, it is a viable scheme for improving the performance of CVQKD by means of these operations [11, 12, 30–33]. Recently, the photon subtraction, as the one of non-Gaussian operations, has been employed extensively for lengthening the maximal transmission distance of one-way CVQKD [11, 12, 30, 31], two-way CVQKD [32] and even measurement device-independent(MDI) CVQKD [33]. In particular, when the so-called virtual photon subtraction was first presented in [12], such an operation not only can be used to simulate the ideal photon subtraction, but also can be implemented via non-Gaussian post-selection, which is applied to extend the maximal transmission distance of SR-CVQKD [23]. Despite the above merits, the success probability of implementing this photon-subtraction operation is usually no more than 0.25 for the variance of the Einstein-Podolsky-Rosen (EPR) state $V = 20$ , which may give birth to the lost of the information between Alice and Bob in the process of extracting the secret key [15]. Attractively, the quantum photon catalysis [34–37], as the other non-Gaussian operation, has been used for the CVQKD system so far, performing better than the single photon subtraction case in terms of the transmission distance and the tolerable excess noise without taking detector efficiency into account [15]. In particular, the zero-photon catalysis operation has the best performance. However, from a practical standpoint, the preparation of EPR state is relatively difficult, which makes it impractical to improve the performance of CVQKD system in the case of bilateral catalysis. On the other hand, during the process of multiphoton catalysis, the preparation of multiphoton sources and the use of multiphoton detections are required, thereby resulting in incompatibility with the telecommunication technique. Therefore, inspired by the above analysis, we suggest a SR-CVQKD scheme with the zero-photon catalysis operation involved an on-off type detector.

In this paper, we focus on the performance improvement of the SR-CVQKD system with the zero-photon catalysis. This paper is structured as follows. In Sec.II, we suggest the conventional SR-CVQKD from the perspectives of the prepare-and-measure (PM) scheme and the entanglement-based (EB) scheme, and then propose the model of zero-photon catalysis (ZPC)-based SR-CVQKD in a realistic detection scenario. The secret key rate of the ZPC-based SR-CVQKD and the performance analysis are provided in Sec. III. Finally, we draw conclusions in Sec. IV.

2. The ZPC-based SR-CVQKD System

In this section, we propose the model design of the SR-CVQKD system involving the zero-photon catalysis operation. From a practical point of view, the PM scheme of SR-CVQKD systems is advantageous to be realized, and its security analysis can be formulated by the equivalent EB scheme. Therefore, we first review the PM and EB scheme of SR-CVQKD, and then detail the SR-CVQKD system with zero-photon catalysis.

Fig. 1 Schematic diagram of the SR-CVQKD system where Alice successively sends weak quantum signal (green pulse) and bright reference (orange pulse) pulses to Bob over an optical channel characterized by channel transmittance T and excess noise ε. At reception, each received pulse is performed by coherent detection in Bob’s own phase reference frame using the LO pulses (blue pulse). GM: Gaussian modulation; T: channel transmittance; ε: excess noise.

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2.1. PM and EB description of SR-CVQKD

For the conventional PM scheme of the SR-CVQKD system, as shown in Fig. 1, Alice sends a coherent state $| q_{A} + i p_{A} 〉$ acting as the quantum signal pulse with two random numbers q_A and p_A from the Gaussian distribution of variance V_A to Bob. In the next time bin, she sends another coherent state $| q_{A R} + i p_{A R} 〉$ as the reference pulse whose fixed amplitude $\sqrt{V_{R}} = \sqrt{q_{A R}^{2} + p_{A R}^{2}}$ is several times greater than $\sqrt{V_{A}}$ but much weaker than the amplitude of the traditional LO. Note that in oder to carry information on Alice’s reference frame, the mean quadrature values of the reference pulse are publicly known. On the receiver side, Bob performs consecutively coherent detections of quantum signal and reference pulses in his own reference frame where a second laser to generate local LO pulses is placed at Bob station so as to estimate the relative phase ${\hat{θ}}_{E}$ . After undergoing the phase correlation on Alice and Bob’s data, a string of secret key finally can be extracted.

However, in the SR-CVQKD system, a critical problem about the estimation of the relative phase ${\hat{θ}}_{E}$ need to be solved, which has an effect on the reference frame alignment. Fortunately, following the approach in [24, 25], Bob performs a homodyne detection on the incoming quantum signal pulse and then takes a heterodyne detection on the incoming reference pulses (including the reference pulse by Alice and the LO by Bob). Thus Bob in his own reference frame can obtain one of the mean quadrature values (q_B or p_B) of the signal pulse and both of mean quadrature values (q_BR and q_BR) of the reference pulse. Moreover, at Alice and Bob’s station, we assume that the two lasers have a stable line width so that the θ_E in each round maintains a specific constant. Under these conditions, the correlation between (q_AR, p_AR) and (q_BR, p_BR) can be established to estimate the relative phase ${\hat{θ}}_{E}$ via

(\begin{matrix} q_{B R} \\ p_{B R} \end{matrix}) = \sqrt{T_{e c t}} (\begin{matrix} cos {\hat{θ}}_{E} & - sin {\hat{θ}}_{E} \\ sin {\hat{θ}}_{E} & cos {\hat{θ}}_{E} \end{matrix}) (\begin{matrix} q_{A R} \\ p_{A R} \end{matrix}),

where

T_{e c t}

denotes the effective channel transmittance. In addition, without loss of generality, we assume that Alice’s p_AR quadrature of the reference pulse is set to zero. Therefore, according to Eq. (1), it is straightforward to derive the relative phase

{\hat{θ}}_{E}

, i.e.,

{\hat{θ}}_{E} = arctan \frac{p_{B R}}{q_{B R}} .

It is evident that the second laser to generate local LO pulses placed at Bob station is just to estimate the relative phase ${\hat{θ}}_{E}$ , which is fully trusted. Unfortunately, in practice, the amplitude of the reference pulse is relatively weaker when compared to the typical LO, so we have to take its quantum uncertainty into account. Thus, the relative phase ${\hat{θ}}_{E}$ should be modified as ${\hat{θ}}_{E} = θ_{E} + ϕ_{e r r}$ with the actual relative phase θ_E and the estimation error $ϕ_{e r r}$ . Note that $ϕ_{e r r}$ is a crucial factor in determining the secret key rate of SR-CVQKD, which satisfies a random probability distribution $P (ϕ_{e r r}) .$ That is because after Bob performs the phase correlation, the remaining phase noise $V_{e s t}$ reads

V_{e s t} = V_{e r r} + V_{c h} + V_{d r i},

where

V_{e r r}

, V_ch and

V_{d r i}

represent the variance of the phase reference’s estimation error, the variance of the channel and the variance of the relative phase drift, respectively. Thus, the phase noise

ξ_{p h a s e}

can be given by

ξ_{p h a s e} = 2 V_{A} (1 - e^{- V_{e s t} / 2}) .

The PM scheme is often implemented in practice systems, while the EB scheme is beneficial to facilitate security analysis, so we have to introduce an equivalent EB version in SR-CVQKD protocol where Alice prepares an EPR source with a modulation variance $V = V_{A} + 1$ and retains mode A to heterodyne detection while sending the other mode B to Bob through the quantum channel characterized by the quantum channel transmittance T and the excess noise ε. Thus, the covariance matrix $Γ_{A B}$ shared between Alice and Bob before executing any detection can be given by

Γ_{A B} = (\begin{matrix} V I I & \sqrt{T μ (V^{2} - 1)} \bar{φ} \\ \sqrt{T μ (V^{2} - 1)} \bar{φ} & T μ (V + χ) I I \end{matrix}),

with

\bar{φ} = (\begin{matrix} \bar{cos ϕ_{e r r}} & \bar{sin ϕ_{e r r}} \\ \bar{sin ϕ_{e r r}} & - \bar{cos ϕ_{e r r}} \end{matrix}),

where

\begin{array}{l} \bar{cos ϕ_{e r r}} = \int_{- π}^{π} d ϕ_{e r r} P (ϕ_{e r r}) cos ϕ_{e r r}, \\ \bar{\sin ϕ_{e r r}} = \int_{- π}^{π} d ϕ_{e r r} P (ϕ_{e r r}) \sin ϕ_{e r r}, \end{array}

If the probability distribution $P (ϕ_{e r r})$ is symmetric around $ϕ_{e r r} = 0$ , the parameter $\bar{φ}$ can be rewritten as $\bar{φ} = \bar{\cos ϕ_{e r r}} σ_{z}$ with $σ_{z} =$ diag $(1, - 1)$ [24]. Consequently, we find that in this sense, the contribution of the reference frame alignment is to rescale the off-diagonal elements of the covariance matrix between Alice and Bob.

2.2. Photon catalysis in SR-CVQKD

To remove the relative phase drift noise of the SR-CVQKD scheme, a method of a balanced delay line interferometer was proposed in [28]. Motivated by this delay line, we propose a SR-CVQKD scheme involving the zero-photon catalysis operation, as shown in Fig. 2. Note that the vacuum state ${| 0}_{c}$ itself on auxiliary mode C remains unaffected but facilitates the transformation of the target ensemble during the zero-photon catalysis (the purple box), so that such operation could effectively prevent the loss of information between Alice and Bob in the process of extracting the secret key. From the perspective of PM version, the ZPC-based SR-CVQKD protocol in Fig. 2(a) is described as follows:

Step 1: Alice generates two coherent states at the repetition rate $f / 2$ consecutively. For each generated coherent state $| Z$ , Alice splits into a weak quantum signal pulse $| Z_{S} 〉$ and a strong reference pulse $| Z_{R} 〉$ by using a balanced delay line interferometer. Note that only the weak quantum signal pulse $| Z_{S} 〉$ is modulated by Gaussian modulation and photon catalysis, transmiting over an optical path of length L_A, while the strong reference pulse $| Z_{R} 〉$ is delayed by a time $1 / f$ on an optical path of length $L_{A} + δ L_{A}$ .
Step 2: At reception, Bob successively generates two coherent states at the same repetition rate $f / 2$ by the second laser. Likewise, by using the same delay line, Bob obtains some coherent LO pulses ( $| γ_{S} 〉$ and $| Z_{R} 〉$ ) where the signal pulse $| γ_{S} 〉$ propagates through an optical path of length L_B, and the reference pulse $| γ_{R} 〉$ is delayed by the same time $1 / f$ on an optical path of length $L_{B} + δ L_{B}$ .
Step 3: Bob uses the incoming reference pulse pairs ( $| Z_{R} 〉$ and $| γ_{R} 〉$ ) to measure the received signal pulses ( $| Z_{S} 〉$ and $| γ_{S} 〉$ ) under his own reference frame by performing coherent detection. More specifically, Bob performs a homodyne detection for the received signal pulse and a heterodyne detection for the reference pulse. According to the measured results, Bob can thus infer an estimation of Alice’s relative phase. Finally, a phase correlation is digitally performed to distill a string of secret key.

Fig. 2 The schematic view of SR-CVQKD protocol with zero-photon catalysis (the purple box) at the sender. (a) Prepare-and-measure scheme of zero-photon catalysis (ZPC) SR-CVQKD. (b). Entanglement-based (EB) scheme of ZPC-based SR-CVQKD. BS: beam splitter; T, ε: channel parameters; ${\hat{Π}}^{o f f}$ : projection operator $| 0 〉〈 0 |$ .

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As shown in Fig. 2(b), it shows the EB scheme of the ZPC-based SR-CVQKD system where Alice generates an EPR state involving two modes A and B with the modulation variance V, which can be expressed as

{| E P R 〉}_{A B} = \sqrt{1 - λ^{2}} exp {λ a^{†} b^{†}} {| 0, 0 〉}_{A B},

with

a^{†} (b^{†})

representing a addition photon operator in mode A(B), and

λ = \sqrt{(V - 1) / (V + 1)}

. In order to be compatible with the existing experimental techniques, during the process of zero-photon catalysis we introduce an on-off type detector illustrated as the projection operator valued measurement (POVM), which is given by

{\hat{Π}}^{o f f} = {| 0 〉}_{c} 〈 0 |, {\hat{Π}}^{o n} = \hat{1} - {| 0 〉}_{c} 〈 0 | .

Note that if the photon number outcome n in mode c is equal to zero (no click), then we identify that the zero-photon catalysis operation is under way. Otherwise, it is considered as the n-photon subtraction $(n > 0)$ . Thus the equivalent zero-photon catalysis ${\hat{O}}_{0}$ can be expressed as

\begin{matrix} {\hat{O}}_{0} = Tr [B (η) {| 0 〉}_{c} 〈 0 |] \\ = : \exp [(\sqrt{η} - 1) b^{†} b] : \\ = {(\sqrt{η})}^{b^{†} b}, \end{matrix}

where

: \cdot :

is a normal ordering prescription,

B (η) = : \exp {(\sqrt{η} - 1) (b^{†} b + c^{†} c) + (b^{†} c - c^{†} b) \sqrt{R}} :

is a beam splitter operator with a transmittance

η = 1 - R

[15, 37]. Interestingly, it is clearly seen from Eq. (9) that the zero-photon catalysis process can be taken as a noiseless attenuation since

{\hat{O}}_{0} | α 〉 \to | \sqrt{η} α 〉

where

| α 〉

is the coherent state in PM scheme of SR-CVQKD.

As mentioned in [15, 34], the effect of quantum catalysis is to facilitate the conversion between modes B and B₁, and hence it results in the state

\begin{array}{l} {| Ψ 〉}_{A B_{1}} = \frac{{\hat{O}}_{0}}{\sqrt{P_{d}}} {| E P R 〉}_{A B} \\ = \frac{\sqrt{1 - λ^{2}}}{\sqrt{P_{d}}} \exp {λ \sqrt{η} a^{†} b^{†}} {| 0, 0 〉}_{A B}, \end{array}

where P_d is the normalized coefficient that represents the success probability for implementing the zero-photon catalysis, and can be derived as

P_{d} = \frac{2}{1 + η + R V},

In Eq. (10), interestingly, the derived state ${| Ψ 〉}_{A B_{1}}$ is still a kind of Gaussian state with a new squeezing parameter $\tilde{λ} = λ \sqrt{η}$ [36, 37]. Thus, the covariance matrix $Γ_{A B_{1}}$ of the state ${| Ψ 〉}_{A B_{1}}$ can be calculated as

Γ_{A B_{1}} = (\begin{matrix} X_{1} I I & Z_{1} \bar{φ} \\ Z_{1} \bar{φ} & Y_{1} I I \end{matrix}),

where

\begin{array}{l} X_{1} = Y_{1} = \frac{2 (1 + V)}{1 + η + R V} - 1, \\ Z_{1} = \frac{2 \sqrt{η (V^{2} - 1)}}{1 + η + R V}, \\ \bar{φ} = \bar{\cos ϕ_{e r r}} σ_{z} . \end{array}

After passing the channel characterized by quantum channel transmittance T and excess noise ε, similar to Eq. (4), the covariance matrix $Γ_{A B_{2}}$ can be calculated as

Γ_{A B_{2}} = (\begin{matrix} X_{1} I I & \sqrt{T_{e c t}} Z_{1} \bar{φ} \\ \sqrt{T_{e c t}} Z_{1} \bar{φ} & T_{e c t} (Y_{1} + χ) I I \end{matrix}),

where the condition of symmetric probability distribution

P (ϕ_{e r r})

is given by

\sin ϕ_{e r r} = 0

, the effective channel transmittance is

T_{e c t} = T μ

, and the channel noise χ is

χ = \frac{1 - T_{e c t} + ϵ_{e l}}{T_{e c t}} + ε,

with the detector electronic noise

ϵ_{e l}

.

3. Performance analysis of the ZPC-based SR-CVQKD system

So far we have derived the parameter characteristics of the SR-CVQKD system involving the zero-photon catalysis operation. In what follows, we demonstrate the effect of the zero-photon catalysis on the SR-CVQKD system in terms of the secret key rate and the maximal transmission distance.

3.1. Asymptotic secret key rate

Now, we consider the secret key rate of our scheme for reverse reconciliation under Gaussian collective attack, where both Alice and Bob perform heterodyne detection and coherent detection (homodyne detection for the received signal pulse and heterodyne detection for the reference pulse), respectively. Fortunately, through the zero-photon catalysis, the yielded state ${| Ψ 〉}_{A B_{1}}$ can still be regarded as a Gaussian state so that we can directly calculate the secret key rate by using the results of the conventional Gaussian CVQKD protocol. According to the optimality of Gaussian attack [38–40], the asymptotic secret key rate K for reverse reconciliation can be obtained by

K = P_{d} {β I (A : B) - I (B : E)},

where P_d has been derived in Eq. (11),

I (A : B)

is the Shannon mutual information between Alice and Bob, and

I (B : E)

is the Holevo bound for Eve’s maximal accessible information. Supposing that

{Q_{A}, P_{A}}

(Q_B or P_B) denotes heterodyne-detection (homodyne detection) results of mode A (B₂),

I (A : B)

can thus be derived as

I (A : B) = \frac{1}{2} \underset{2}{\log} \frac{V_{A}^{^{'}}}{V_{A | B}^{^{'}}},

with

V_{A}^{^{'}} = (X_{1} + 1) / 2

,

V_{B}^{^{'}} = T_{e c t} (X_{1} + χ)

and

V_{A | B}^{^{'}} = V_{A}^{^{'}} - T_{e c t} {(Z_{1} \bar{cos ϕ_{e r r}})}^{2} / 2 V_{B}^{^{'}}

. Moreover, according to Eq. (3), the remaining phase noise

V_{e s t}

for the ZPC-based SR-CVQKD with delay line can be written as [18–20]

V_{e s t} = V_{e r r} = \frac{χ + 1}{V_{R}} + \frac{1}{T μ V_{R}} .

If the probability distribution $P (ϕ_{e r r})$ is tight, we can then obtain the relation as follows

{\bar{cos ϕ_{e r r}}}^{2} = 1 - V_{e r r},

and thus the analytical expression of the mutual information

I (A : B)

can be achieved from Eq. (17). In addition, to solve the Holevo bound, assuming that Eve can purify the whole system, we obtain

\begin{matrix} I (B : E) = S (E) - S (E | B) \\ = S (A B) - S (A | B) \\ = \overset{2}{\sum_{i}} G [(λ_{i} - 1) / 2] - G [(λ_{3} - 1) / 2], \end{matrix}

where

G [x] = (x + 1) \underset{2}{\log} (x + 1) - x \underset{2}{\log} x,

and the symplectic eigenvalues

λ_{i} (i = 1, 2, 3)

are derived as

\begin{array}{l} λ_{1, 2}^{2} = \frac{Δ \pm \sqrt{Δ^{2} - 4 D^{2}}}{2}, \\ λ_{3}^{2} = X_{1}^{2} - \frac{X_{1} Z_{1}^{2} (1 - V_{e r r})}{X_{1} + χ}, \end{array}

with

Δ = X_{1}^{2} + T_{e c t}^{2} {(Y_{1} + χ)}^{2} - 2 T_{e c t} Z_{1}^{2} (1 - V_{e r r})

and

D = X_{1} T_{e c t} (Y_{1} + χ) - T_{e c t} Z_{1}^{2} (1 - V_{e r r})

.

Fig. 3 The maximal secret key rate at each transmission distance with several different parameters ϵ_el ∈ {0.01, 0.001} and V_R ∈ {20V_A, 50V_A } for the optimal choice of η. The black thin line represents the original protocol. The magenta thick line represents the single-photon subtraction. The blue dashed line represents the zero-photon catalysis.

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Fig. 4 The optimal η for the maximal secret key at each transmission distance with several different parameters ϵ_el ∈ {0.01, 0.001} and V_R ∈ {20V_A, 50V_A}. The blue dashed line represents the zero-photon catalysis.

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3.2. Performance analysis

Before demonstrating performance improvement of the ZPC-based SR-CVQKD system, we set the simulation parameters as $V_{A} = 40, β = 0.95, ε = 0.01$ in shot noise units (SNU) for $μ = 0.719$ . Moreover, it is worth noting that the excess noise contributed by the reference pulses (V_R can be set as some realistic values of $20 V_{A}$ and $50 V_{A}$ ) can be neglected when Alice’s amplitude modulator is set to be 60 dB, as expected [28]. As shown in Fig. 3, it shows the maximal secret key rate of the ZPC-based SR-CVQKD system for all possible η with parameter $ϵ_{e l} \in {0.001, 0.01}$ . In Fig. 4 it shows characteristics of the optimal η, which may have effect on the maximal secret key rate in Fig. 3. The black solid line represents the original protocol, which is outperformed by the proposed protocol with zero-photon catalysis operation at long transmission distance range. We find that the zero-photon catalysis operation has advantageous in dramatically enhancing the secret key rate and lengthening the maximal transmission distance. However, at the short-distance range, we find that, for $V_{R} = 50 V_{A}$ , the secret key rate of the ZPC-based SR-CVQKD system is the same as that of the original protocol since its transmittance η = 1 means that there is no quantum catalysis effect (see Fig. 4(b)). Interestingly, although the performance of the proposed protocol for $V_{R} = 20 V_{A}$ is better than that of the original protocol at long-distance range, it seems to be worse than that of the protocol for $V_{R} = 50 V_{A}$ . This indicates that the maximal transmission distance is proportional to the amplitude of the reference pulse [24].

Fig. 5 The maximal tolerable excess noise as a function of transmission distance with several different parameters ϵ_el ∈{0.01,0.001} and V_R ∈ {20V_A, 50V_A}, when optimized over η of Alice’s beam splitter. The black thin line represents the original protocol. The magenta thick line represents the single-photon subtraction. The blue dashed line represents the zero-photon catalysis.

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In addition, the tolerable excess noise is another important factor for evaluating the performance of the ZPC-based SR-CVQKD system. In Fig. 5, it shows the maximal tolerable excess noise as a function of the transmission distance when optimizing over η. The performance of the proposed protocol is better than that of the original protocol, which means that the advantages of zero-photon catalysis can improve the maximal tolerable excess noise even for remote users. For instance, when $ε \sim 0.001$ , the proposed protocol can expand the maximal transmission distance to more than 30 km. To our knowledge, for all the photon-subtraction-based SR-CVQKD schemes in Ref. [23], the single-photon subtraction (SPS)-based SR-CVQKD presents the best performance. Consequently, in order to make comparisons about the performances between the SPS-based SR-CVQKD and the ZPC-based SR-CVQKD, the magenta solid line represents the single-photon subtraction case with respect to the the maximal transmission distance and the maximal tolerable excess noise, as shown in Figs. 3 and 5. It is found that the proposed ZPC-based SR-CVQKD system outperforms the SPS-based SR-CVQKD in terms of the maximal transmission distance and the maximal tolerable excess noise. In particular, at the short-distance range, the performance of the SPS-based SR-CVQKD is exceeded by that of the ZPC-based SR-CVQKD system. The reason is that the zero-photon catalysis has an advantage of the success probability over the single-photon subtraction case (see Fig. 6). These results show that the zero-photon catalysis can be used to improve the performance of SR-CVQKD system.

Fig. 6 The success probability P_d of implementation of zero-photon catalysis as a function of η. As a comparison, the magenta solid line represents the single-photon subtraction case.

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Fig. 7 When V_R = 20V_A and ϵ_el = 0.001, the transmission distance as a function of the estimated parameter η. The color region denotes the secret key rate. The red solid line denotes the optimal value of η corresponding to K_Opt. The orange (black) dashed line denotes the upper (lower) of η, if the secret key rate maintains more than 90%K_Opt. K_Opt: the optimal secret key rate.

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Fig. 8 Performance comparison of SR-CVQKD with the zero-photon catalysis and the single-photon subtraction for different quantum detection efficiency μ and electronic noise ϵ_el of the coherent detections, when given when given V_R = 20V_A and transmission distance 20 km.

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In practice, the parameter η brought by quantum catalysis have an impact on the performance of ZPC-based SR-CVQKD. The reason is that the accurate estimation of η to maintain a relatively high performance usually requires some complicated implementations if the secret key rate changes significantly with η around its optimal value. Fortunately, the secret key rate slowly changes with η at each distance around its optimal value $K_{O p t}$ (red solid line), as depicted in Fig. 7. We find that the secret key rate can maintain higher than 90% of its optimal value ( $K_{O p t}$ ) when the estimated η locates in the region between the black dashed line and the orange dashed line. On the other hand, As shown in Figs. 3 and 5, the imperfection of coherent detections is also an important factor that can interfere with the performance of ZPC-based SR-CVQKD system. Fig. 8 shows the performance comparison of SR-CVQKD with the zero-photon catalysis and the single-photon subtraction for different quantum detection efficiency μ and electronic noise $ϵ_{e l}$ of the coherent detections, when given $V_{R} = 20 V_{A}$ and transmission distance 20 km. We can find that the performance of the ZPC-based SR-CVQKD always is better than that of the SPS-based SR-CVQKD when μ and $ϵ_{e l}$ take a finite value. In other words, the SR-CVQKD system with the zero-photon catalysis requires lower quantum detection efficiency μ and higher electronic noise $ϵ_{e l}$ than the SPS-based SR-CVQKD under the case of achieving the same performance.

4. Conclusions

We have suggested a method to improve the performance of the SR-CVQKD system by using the zero-photon catalysis, which can be taken as a noiseless attenuation. Moreover, we derive the asymptotic secret key rate of the ZPC-based SR-CVQKD system for reverse reconciliation against the collective attack, according to the optimality of Gaussian attack. The numerical simulations show that comparing with the original protocol, the ZPC-based SR-CVQKD system has an advantage of lengthening the maximal transmission distance with the increased secret key rate. Attractively, in order to further highlight the advantage of applying the zero-photon catalysis in the SR-CVQKD systems, we make a comparison of the SR-CVQKD schemes involving zero-photon catalysis and single-photon subtraction, and find that the performance of the ZPC-based SR-CVQKD system is superior to that of the single-photon subtraction case in terms of the transmission distance and the tolerable excess noise. In particular, the ZPC-based SR-CVQKD system allows the lower quantum detection efficiency and the higher electronic noise to achieve the same performance.

Funding

National Natural Science Foundation of China (61572529, 61821407, 11664017); Outstanding Young Talent Program of Jianxi Province (20171BCB23034).

Acknowledgments

LiYun Hu is supported by the Outstanding Young Talent Program of Jianxi Province (20171BCB23034).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Improvement of self-referenced continuous-variable quantum key distribution with quantum photon catalysis

Abstract

1. Introduction

2. The ZPC-based SR-CVQKD System

2.1. PM and EB description of SR-CVQKD

2.2. Photon catalysis in SR-CVQKD

3. Performance analysis of the ZPC-based SR-CVQKD system

3.1. Asymptotic secret key rate

3.2. Performance analysis

4. Conclusions

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (8)

Equations (21)

Optics Express