Quantum key distribution (QKD) allows two remote parties, Alice and Bob, to distill a secret key over insecure quantum and classical channels.^[1–4] There are two main versions of QKD, i.e., discrete-variable (DV) QKD^[5–7] and continuous-variable (CV) QKD.^[8–10] DV-QKD protocols encode information on properties of single photons, and use single-photon counters to measure the photon number. While CV-QKD protocols encode information on the quadrature variables of the optical field, and use a homodyne or heterodyne detector to measure the field quadratures.

A CV-QKD protocol based on Gaussian-modulated coherent states has been proved to be a practical scheme.^[11,12] However, the level of signal-to-noise ratio (SNR) has an important effect on the reconciliation efficiency of this protocol, namely, we can obtain high reconciliation efficiency when the protocol operated at high SNR and achieve low reconciliation efficiency when the protocol operated at low SNR values.^[13] Fortunately, discrete modulation CV-QKD protocol has no such limitation, and even at low SNR values, this kind of protocol can still have a high reconciliation efficiency. In this paper, we mainly study the four-state CV-QKD protocol, which has been proved to be secure against collective attacks^[14–17] and has been experimentally demonstrated.^[18]

In order to improve the performance and balance the secret key generation rate and the maximum secure distance of four-state CV-QKD protocol, we insert a linear optics cloning machine (LOCM)^[19,20] inside Bob’s apparatus. The LOCM, in fact, is the cloning machine which can be practically implemented using just linear optics, homodyne measurement and controlled displacements.^[21] It is remarkable that the LOCM used here does not violate the non cloning theorem since we make approximate copies instead of perfect copies of quantum states in our protocol.^[22] The LOCM operation provides a method to control the useful noise by tuning its parameters in appropriate ranges.^[19] Therefore, a good performance of four-state CV-QKD protocol can be achieved by using the LOCM operation.

The paper is organized as follows. In section 2, we give our theoretical analysis of the four-state protocol. In section 3, a schematic configuration of the four-state CV-QKD protocol based on LOCM operation is described in detail. In section 4, we derive the secret key generation rate of the four-state CV-QKD protocol based on LOCM operation under collective attacks. We compare the performance of the four-state CV-QKD protocol based on LOCM operation with the standard four-state CV-QKD protocol in section 5. The conclusions are drawn in section 6.

We introduce the four-state protocol based on two versions. One is the prepare-and-measure (PM) version, and the other is the entanglement-based (EB) version. Subsequently, we give the region in which the eavesdropper’s (Eve) mutual information with reference party (Bob) is similar in the four-state CV-QKD protocol and the Gaussian CV-QKD protocol.

2.2. Covariance matrix of the four-state protocol

The initial covariance matrix of the state is given by^[13] 5 where I and σ_z are, respectively, the identity matrix and the usual Pauli matrix; V is the variance of the trusted modes (A and B) in the quadratures which can be written as . The parameter 6 refers to the correlation between the trusted modes. After mode B passes through a quantum channel, the state is transformed into with covariance matrix 7 where is the channel-added noise referred to the channel input, T is the channel transmittance and ε is the excess noise. Note that in the Gaussian modulation protocol, this covariance matrix can be reconstructed by replacing Z₄ with Z_G, where . Fortunately, Z₄ is very close to Z_G when V_A is small enough, as shown in Fig. 1. In other words, for , Z₄ and Z_G are almost equal, meaning that in this region, Eve’s mutual information with Bob is very similar in four-state protocol and Gaussian modulation protocol.^[24]

Fig. 1.

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	Fig. 1. (color online) Comparison of the correlation for two different modulation protocols. Full lines correspond to the four-state protocol (correlation marked as Z4) and dashed lines correspond to the Gaussian modulation protocol (correlation marked as ZG). Both Z4 and ZG are functions of the modulation variance . (a) VA in a large range, and (b) VA in a small range.

In this section, a four-state CV-QKD protocol based on LOCM operation is proposed, as shown in Fig. 2. It is worth noting that we use the LOCM to make the approximate copies instead of perfect copies of quantum states. As we all know, the non cloning theorem forbids making perfect cloning, consequently, the LOCM used here does not violate the non cloning theorem.^[22] In this protocol, Alice applies the heterodyne detection, and obtains two measurements and . After receiving the coherent states centered at , , Bob prepares for two vacuum states and , then the LOCM operation is implemented, and the quadratures are measured by Bob via performing either homodyne or heterodyne detection. In the LOCM operation, due to the symmetry of all transformations on pairs of quadratures , without loss of generality, only one quadrature is considered here. On one hand, at Bob’s side, a beam splitter (BS) with transmissions τ reflects a fraction of the input mode , as displayed on Fig. 2(a), then we obtain 8 and subsequently 9

Fig. 2.

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	Fig. 2. The EB scheme for the four-state CV-QKD protocol based on LOCM operation. (a) The four-state CV-QKD with LOCM in terms of two parameters λ and τ. (b) The equivalent four-state CV-QKD protocol with a thermal state of variance characterized by the tunable noise and transmittance .

On the other hand, the transmitted mode can be written as 10 then based on the measurement result with the tuning gain λ, the transmitted mode can be displaced as 11 Based on the right side of Eq. (11), the variance of the mode can be calculated as 12 According to Eq. (12), the expression of transmittance is given by 13 With the constraint in mind, it is necessary for us to solve the parameter λ in a range for the given parameter . Thus, we can derive the LOCM-tuned noise with two parameters λ and τ referred to Bob’s input, the expression can be written as 14

The equivalent system can be established by combining a thermal state of variance on a beamsplitter of transmittance , as shown in Fig. 2(b).

It is clear that is a function of two parameters λ, τ, as shown in Fig. 3. We can regulate the in a range [0,1] by properly tuning these two parameters.

Fig. 3.

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	Fig. 3. (color online) Relation between the LOCM-tuned transmittance and two parameters λ and τ. It is clear that the transmittance can be regulated in [0,1] by choosing some appropriate parameters λ and τ.

Note that when we consider an additional constraint on parameters λ and τ of LOCM operation,^[25] equation (11) can be rewritten as 15 If we set , obtaining , which is an exact operation of a linear optics amplifier (LOA) on the input mode of Bob. According to Ref. [13], the LOA is a phase-insensitive amplifier with gain . The variance of the quadrature can be calculated as 16 Using a method similar to the one mentioned above, we can derive a quantum channel which features a transmittance in the LOA quantum system. As a result, the LOA-tuned noise can be calculated as and this noise can be regulated in a simple range since . Comparing the tunable noise with χ_a, it is obvious that we can tune two parameters λ and τ of but only one parameter τ of χ_a. Thus, the LOCM-tuned noise characterized by two parameters λ and τ can be made much larger than the LOA-tuned noise χ_a, which is dependent on only one parameter τ in a large range.

To describe the relation between the tuned noise based on LOCM operation and the tuned noise χ_a based on LOA operation, we particularly define the differential noise , as shown in Fig. 4. This parameter contains the information of both LOCM-tuned and LOA-tuned noise. If we set , meaning that the tuned noise based on LOCM operation is exactly equivalent to the tuned noise based on LOA operation. In other words, the LOCM can be transformed into LOA for .

Fig. 4.

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	Fig. 4. (color online) Relation between the differential noise and two parameters λ, τ. We can obtain easily and then derive by selecting appropriate parameters λ and τ.

Note that the homodyne or heterodyne detection at Bob’s side may be followed by a noise with variance (implementation of LOCM operation) or χ_a (implementation of LOA operation) in the four-state CV-QKD protocol. Under certain conditions ( ), these two operations are virtually equivalent which has been demonstrated above. Once this implementation procedure is ended, data postprocessing is conducted by Alice and Bob, and then the final secret key is generated. Here we should remark that in direct reconciliation, the data owned by Alice is seen as a reference to complete the key establishment and in reverse reconciliation, the data owned by Bob is the reference.^[26] Reverse reconciliation has been demonstrated that it can offer a great advantage in performance improvement of the CV-QKD,^[11,13] thus we perform our calculations with this reconciliation.

Based on the above-mentioned analysis, we will limit our calculations to reverse reconciliation. The secret key generation rate for reverse reconciliation under collective attacks can be given by 17 where and β are, respectively, the mutual information of Alice and Bob and reconciliation efficiency; is Eve’s mutual information with Bob.

In order to predigest the security analysis of the four-state CV-QKD protocol based on LOCM operation, we suppose that Alice applies heterodyne detection with measurements . Considering the fact that the degraded LOCM operation is essentially a phase-insensitive amplifier, therefore, Bob here performs heterodyne detection via using a detector, which features an efficiency η and an electronics noise .^[13] For simplicity, we assume the heterodyne detector used by Bob is perfect in our protocol, which means η = 1 and .^[13,25] It is remarkable that the four-state CV-QKD protocol with LOCM is equivalent to a system by combining a thermal state of variance , which are shown in Figs. 2(a) and 2(b). Therefore, the additional noise referred to the input of the detector can be derived as based on Fig. 2(b).^[25]

For the protocol based on LOCM operation, the mutual information of Alice and Bob can be calculated from the measurement variance of Bob and conditional variance which is given by^[12] 18 where represent the total noise referred to the channel input.

We have mentioned in the previous that when , Eve’s mutual information with Bob is similar in four-state protocol and Gaussian modulation protocol. Therefore, one has . Based on this result, the expression of can be given by 19 where is the von Neumann entropy of the state ρ. Considering the fact that Eve can purify the system EAB₁ and Bob’s measurement can purify the system AEFG [the mode notations are shown in Fig. 2(b)], we thus get and . Then can be rewritten as 20 For the first term , we can use the symplectic eigenvalues of covariance matrix to calculate it. The symplectic eigenvalues can be expressed as^[13] 21 with 22

In order to calculate , we need to derive the symplectic eigenvalues of the covariance matrix characterizing the state . The matrix can be written as 23 where and MP is the Moore–Penrose inverse of a matrix. The decomposed covariance matrix is given by 24 It is clear that we can achieve the sub-matrices γ_AFG, , and from this matrix. Besides, matrix can be obtained from the rearrangements of lines and columns of the matrix characterizing the system : 25 where represents the beamsplitter transformation performed on modes B and F₀ with26 and the matrix illustrates an effect of the additional LOCM-tuned noise which can be written as 27 After that, we can calculate the eigenvalues , which are given by^[24] 28 where for the heterodyne detection case, 29

Based on these consequences, the expression for can be written as 30 where .

Considering the fact that the four-state protocol based on LOCM operation depends much on parameters λ and τ, thus we can tune λ and τ of the LOCM operation in appropriate ranges to improve the performance of the four-state protocol. We have mentioned in the above that the LOCM-tuned noise characterized by two parameters λ and τ can be made much larger than the LOA-tuned noise χ_a, which is dependent on only one parameter τ in a large range. Therefore, in this section, we mainly study the effect of the LOCM operation on the performance of four-state CV-QKD protocol for reverse reconciliation with the heterodyne detection, as the LOA operation can be analyzed in a similar way.

In order to simplify the analysis, as mentioned above, we assume that Bob uses a perfect heterodyne detector in our protocol as well as in the standard four-state protocol, namely, the parameters η = 1 and . Other parameters such as reconciliation efficiency β and excess noise ϵ are fixed in all simulations to the values ϵ = 0.005 and β = 0.8,^[24] respectively. The channel transmittance is , where /km is the loss coefficient of optical fibers and L is the channel length. Based on the previous analysis, correlations Z₄ and Z_G are almost equal when , thus we can take and in our simulations.

In what follows, we compare the performance of the four-state CV-QKD protocol based on LOCM operation with the standard four-state CV-QKD protocol, as shown in Fig. 5 with gain λ = 0.3. We take τ = {0.2,0.4,0.6} as examples, the corresponding calculated from Eq. (13) are , respectively, which are all physical results. In Fig. 5, for the gain value λ = 0.3, the secret key generation rates of our protocol are higher than the standard four-state protocol over long distance, and what is more, we can tune the parameter τ to improve the performance of our protocol. It is clear in Fig. 5 that we can achieve the higher secret key rate and longer maximum secure distance by tuning the value of parameter τ to be smaller in an appropriate range.

Fig. 5.

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	Fig. 5. (color online) Comparison of the performance between the four-state protocol based on LOCM operation and the standard four-state protocol with parameters λ = 0.3 and τ = {0.2,0.4,0.6}.

For simplicity, we take the fixed efficiency τ = 0.4 for example to illustrate the effect of the tuned gain on the four-state protocol based on the LOCM operation in Fig. 6. Here, we choose λ = {0.1,0.3,0.5}, thus the corresponding calculated from Eq. (13) are {0.6713,0.8258,0.9964} respectively, which are all physical results. As shown in Fig. 6, the secret key generation rates of our protocol are higher than the standard four-state protocol over a long distance. Besides, we can tune the parameter λ in this case to improve the performance of our protocol. It is clear in Fig. 6 that we can achieve the higher secret key generation rate and longer maximum secure distance by tuning the value of parameter λ to be larger in a suitable range.

Fig. 6.

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	Fig. 6. (color online) Comparison of the performance between the four-state protocol based on LOCM operation and the standard four-state protocol with parameters τ = 0.4 and λ = {0.1,0.3,0.5}.

It is necessary to point out that the maximum secure distance of our protocol is slightly shorter than the standard four-state protocol. Fortunately, we can reduce the distance gap between our protocol and the standard four-state protocol by controlling the parameters λ and τ of LOCM operation. For example, in Fig. 5, when we choose λ = 0.3 and τ = 0.2 ( ), the distance gap between our protocol and the standard four-state protocol is about 2 km and in Fig. 6, when we choose τ = 0.4 and λ = 0.5 ( ), the distance gap between our protocol and the standard four-state protocol is about 4 km. In other words, based on LOCM operation, the secret key generation rate of our protocol improves in a large range of distance with the price of decreasing little of the maximum transmission distance compared with the standard four-state protocol.

We have proposed an improved four-state CV-QKD protocol by inserting the tunable LOCM with reverse reconciliation. The LOCM is actually a cloning machine which can be implemented by using linear optics, homodyne measurement and controlled displacement. We have pointed out that the LOCM used here does not violate the non cloning theorem since we make approximate copies instead of perfect copies of quantum states in our protocol. The secret key generation rate can be commendably harmonized with the maximum secure distance as the parameters λ and τ can be stabilized to an appropriate value. We focus on reverse reconciliation in performance analysis mainly because this reconciliation has been proved to provide a great advantage in CV-QKD protocol. Simulation results show that by tuning the parameters of LOCM operation in a suitable range, we can improve the secret key generation rate of the four-state protocol over long distance and decrease the distance gap between our protocol and the standard four-state protocol.