Finite-size analysis of unidimensional continuous-variable quantum key distribution under realistic conditions

: We analyze the unidimensional (UD) continuous-variable quantum key distribution protocol in a finite size scenario under realistic conditions. The dependence of the secret key rate on realistic parameters is analyzed numerically. A method of calculating the optimal ratio to divide the data samples in order to achieve the largest secret key rate is proposed. When the data samples are large, the superiority of the UD protocol in data processing becomes apparent. It is expected that the features and methods presented in this paper will aid in the exploration of the latent capacity of the UD protocol as well as the development of further applications


Introduction
Quantum key distribution (QKD) allows two remote and legal parties, Alice and Bob, to share a common secret key in a quantum channel, which is assumed to be controlled by a potential eavesdropper Eve, and an authenticated classical channel [1,2].As a safe and advanced communication method, its security is based on the basic principles of quantum mechanics.
Unlike discrete-variable QKD, continuous-variable (CV) QKD encodes the information into continuous-spectrum quantum observables and utilizes homodyne detectors instead of singlephoton detectors.During the past decade, the theory and technology of CV QKD have undergone rapid development .It is known that CV QKD protocols based on nonclassical states demonstrate better performance than those based on coherent states.However, as the coherent state is easier to generate and only requires classical telecom technologies, coherent state based protocols have developed rapidly.Prototype machines have been developed, and several field tests have been completed [29][30][31].It is believed that protocols based on coherent states can be used in practical applications in near future.
A further simplified unidimensional (UD) CV QKD protocol based on coherent states has recently been proposed [32].Nearly simultaneously to this development, UD CV QKD was realized experimentally [33,34].This asymmetric UD coherent state protocol allows the sender, Alice, to use one modulator instead of two, thereby reducing the complexity and cost of Alice's apparatus.However, all previous works failed to take into account the finite-size effects, which are critical to the practical security of the QKD system.
In this paper, we mainly analyze the UD CV QKD protocol in a finite size scenario under realistic conditions.Firstly, in order to lay a good foundation, an introduction to the UD protocol under realistic conditions is presented.The analytic expression for the symplectic eigenvalues, which are used to calculate the secret key rate, is derived in order to significantly reduce the calculation time.Next, the dependence of the asymptotic secret key rate on realistic parameters, which will provide a useful guide for parameter selection, is investigated.When the amount of data samples used for parameter estimation is large in the finite size scenario, the secret key rate will approach the asymptotic secret key rate.
Unlike with the symmetrical (SY) coherent state protocol [7,13], in the UD protocol we must estimate the transmission efficiency and excess noise of one quadrature, as well as the variance of the other quadrature.It is well known that the larger the amount of data samples used for parameter estimation, the weaker the finite size effect [35].However, taking into consideration the real-time and stability of the QKD system, taking a longer period of time to increase the total number of samples is not necessarily beneficial.In order to distill the largest secret key rate from a limited total number of samples, a method is proposed to subdivide the data samples with an optimal ratio.After comparison, we can see that the UD protocol can achieve similar performance to its SY counterpart under realistic conditions, when the excess noise is low and number of samples is large.In particular, when the amount of data samples is large, the superiority of the UD protocol in data processing emerges.
In Sec.II, the UD CV QKD protocol under realistic conditions is presented, and the security under collective attack is analyzed using the entanglement-based (EB) scheme.Section III provides the numerical analysis of the secret key rate, depending on realistic parameters such as the reconciliation efficiency, detection efficiency, and electronic noise.Section IV presents the method of calculating the optimal ratio to divide the data samples in order to achieve the largest secret key rate under Gaussian collective attack in a finite size scenario.Finally, Section V presents the conclusions.

An introduction to unidimensional protocol schemes
The prepare-and-measure (PM) scheme is shown in Fig. 1(a).The sender, Alice, displaces independent coherent states produced by a laser to a Gaussian distribution in the amplitude or phase quadrature, using one amplitude or phase modulator with a modulation variance M V .Note that the variances in this paper are all normalized to shot noise units.The Gaussianmodulated quantum states form a unidimensional chain structure in phase space with a length of 1 M V + and a thickness of 1. Alice sends these quantum states to Bob through an untrusted quantum channel, characterized by its transmission , x y T T and excess noise , x y ε ε .
On the receiver Bob's side, a balanced homodyne detector (BHD) with efficiency η and electronic noise e υ is used to measure the amplitude or phase quadrature.In realistic conditions, it is supposed that the eavesdropper Eve cannot access Bob's apparatus.The efficiency η and electronic noise e υ are phase insensitive and should be calibrated before the communication.The EB scheme, which is equivalent to the PM scheme of the UD protocol, is shown in Fig. 1(b).On Alice's side, an Einstein-Podolsky-Rosen (EPR) state 0 AB ρ with variance V is utilized.Then, Alice squeezes one of its modes B0 with squeezing parameter ln r V = . We denote the output mode S. Without loss of generality, we assume that the phase quadrature is squeezed.The resulting covariance matrix AS γ is The mode S , with variance 2 +1 in the amplitude quadrature, is sent to the remote trusted party, Bob, through a phase-sensitive channel characterized by the transmission , ε ε ; the covariance matrix then becomes ( ) where 1

B y
V is the variance of mode 1 B in the phase quadrature, 1 y B C is the correlation between the two modes A and 1 B in the phase quadrature, and linex χ is the channel noise in the amplitude quadrature and can be expressed as ( ) Because there is no modulation in the phase quadrature, the correlation 1 y B C cannot be estimated.
In the EB scheme, the realistic BHD can be modeled as a beam splitter with transmission η and a perfect BHD.The electronic noise e V of the realistic BHD can be modeled as a thermal state 0 R ρ , which could be considered to be the reduced state obtained from an EPR

Security analysis
In the asymptotic limit, the collective attack has been proven to be the optimal attack [14] and the corresponding secret key rate is given by , where AB I is the Shannon mutual information between Alice and Bob, BE χ represents the Holevo bound between Bob and Eve for reverse reconciliation, and β is the reverse reconciliation efficiency.The mutual information can be calculated by the following expression The Holevo bound is defined as ( ) where ( ) After an elaborate derivation, the analytical expression for the symplectic eigenvalues of the covariance matrix 1 AB γ is ( ) where ( ) ( ) The analytical expressions for the symplectic eigenvalues 3,4,5 λ of the covariance matrix ) , 1 ) The Holevo bound can be obtained by ( ) ( ) -( ) where ( ) ( ) ( ) where We then obtain the parabolic equation where ( ) The parabolic curve between 1

B y
C and 1

B y
V is shown in black in Fig. 2. The whole plane is divided into two regions by the parabolic curve, i.e., physical region and unphysical region.The region contained by the parabolic curve is the physical region, which is divided into secure and unsecure regions by the solid cyan curve.In the secure region, the secret key rate is greater than zero.In the unsecure region, the secret key rate is less than zero.In the From the above numerical analysis, it is evident that in order to achieve a higher secret key rate, a higher reconciliation efficiency, higher detection efficiency, and lower electronic noise are required.Currently, the highest reconciliation efficiency that can be achieved is 97% [24], and the detection efficiency in a fiber-based system is ~0.6.The typical value of the electronic noise of 0.1 is selected.
With the parameters 0.97

Secret key rate in the finite size scenario
In a practical UD CV QKD protocol, the unknown parameters are estimated using finite-sized data samples.In this section, we consider the Gaussian collective attack and analyze the secret key rate in the finite-size scenario.
The secret key rate considering the finite-size effect can be written as [35,38,39] ( ( ) , ) where where  is the probability of error during privacy amplification; a conservative value of -10 =10  is utilized here.

Parameter estimation
In the UD protocol, Alice modulates the coherent states in the amplitude quadrature, and Bob measures the transmitted coherent states in the same quadrature.At the same time, Bob also needs to randomly switch the detection bases to measure the phase quadrature.Alice then rejects the portion of her data corresponding to the switched bases when Bob makes public his detection bases.After this process, Alice and Bob can share two groups of correlated data samples ( ) , where A x ( B x ) represents Alice's data (Bob's data) in amplitude quadrature, and l represents the number of Bob's data samples in phase quadrature.The l phase quadrature data will be used to evaluate the variance of the phase quadrature B y V .From the N l − amplitude quadrature data, Alice and Bob will select m data samples to evaluate the transmission efficiency x T and excess noise x ε .
The data obtained in the amplitude quadrature is linked through the following linear equation: , To calculate the final secret key rate using Eq. ( 19), the estimated values of ˆx T , ˆx ε , and 1 ˆB y V are substituted by their expected values ( ) and ( ) Here we assume that 1 1 , y x T T = , and y x ε ε = .

Determining the largest secret key rate among N total samples
It is well known that the larger the number of data samples used for parameter estimation, the less the finite-size effects.Considering the real-time stability of the CV QKD system, it is not necessarily beneficial to increase the total number of samples.Given a number of total samples N , there is a trade-off to consider in assigning them.The proportion of the total samples used for parameter evaluation should be optimized to maximize the secret key rate.
In the SY coherent state protocol, when r samples are selected to evaluate the parameters, only the proportion / r N needs to be determined.However, in the UD protocol there are two parameters, m and l .Thus, not only the proportion / m N , but also the proportion / l N needs to be determined.In order to determine the largest secret key rate with N total samples, a method was designed to scan the proportions of / m N and / l N at the same time to achieve an optimal proportion ( ) : : m l n .The scanning result can be seen in Fig. 8.
In Fig. 8, we can see that there exists a best proportional (BP) point (red diamond) corresponding to the largest secret key rate.The coordinate of the BP point is ( / l N , / m N , BP m K ) = (0.19, 0.155, 2.2 × 10-3), where BP m K is the secret key rate of the BP point, i.e., the maximum secret key rate that can be achieved when the total sample is limited.In Fig. 8, the total data sample is 9 10 N = and the optimal ratio is (0.19: 0.155: 0.655).Thus, the number of samples used to evaluate the parameters x T and x ε is protocol.Comparing the secret key rates for different total samples, we can see that the secret key rate f m K decreases more rapidly than f K as the total samples decrease.Therefore, the UD protocol is more sensitive to the total number of samples than the SY protocol.In the SY protocol, the number of random numbers used for detection base switching is a constant, N .In the UD protocol, the proportion l N decreases with the increasing total number of samples.This means that a smaller amount of random numbers was required to switch the detection bases to measure the non-modulated phase quadrature.Furthermore, the asymmetrical base switching in the UD protocol decreases the amount of information required for the public declaration of Bob's measurement bases and facilitates Alice's data sifting.Therefore, the superiority of the UD protocol emerges for QKD systems with large data samples [40].

Conclusion
In this paper, the finite size effect of the UD continuous-variable quantum key distribution protocol is analyzed under realistic conditions.We believe that the characteristics discovered and methods proposed in this paper will aid in the exploration of the latent capacity of the unidimensional protocol, as well as the discovery of more applications sensitive to the cost of the quantum key distribution system.The composable security [41,42] of the UD protocol will be considered in a further theoretical analysis.

Fig. 1 .
Fig. 1.PM and EB schemes of the UD protocol under realistic conditions.
N V entering the other input port of the beam splitter.The variance N V has a relationship with e υ given by ( ) beam splitter, the covariance matrix AB γ has the form total noise added between Alice and Bob relative to the channel input in the amplitude quadrature, and ( ) noise introduced by the realistic BHD relative to Bob's input in the amplitude quadrature.B y V is the variance of mode B in the phase quadrature.The noise hom χ , which is phase insensitive, has the same value in both the amplitude and phase quadrature.y B C is the correlation between the two modes A and B in the phase quadrature.

S
ρ is the von Neumann entropy of the eavesdropper's quantum state E

Fig. 6 .
Fig. 6.(a) Secret key rate m K ∞ as a function of the phase quadrature variance 1 B y V with secret key rate m K ∞ as a function of the distance is presented with different excess noise values in Fig.7.It should be noted that the modulation variance is optimized at each distance.Compared to its SY counterpart, the UD protocol is more sensitive to the excess noise.When the excess noise is low, a comparable secret key rate and transmission distance can be achieved.

Fig. 7 .
Fig. 7. Secret key rate versus distance in the SY and UD coherent state protocols for different excess noise values.
, η and e υ can be well-calibrated on Bob's side.The modulation variance of the random Gaussian variable Ax is M V .In the symmetrical CV QKD protocol, it has been proven that the information gained by Eve, is the total number of signals exchanged between Alice and Bob, in which n scales the number of signals used to extract the secret keys, and N n − scales the number of the remainder of the signals for parameter estimation.