# Noiseless Linear Amplifiers in Entanglement-Based Continuous-Variable Quantum Key Distribution

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## Abstract

**:**

## 1. Introduction

## 2. Entanglement-Based CV-QKD Protocols

#### 2.1. Entanglement Distribution: Entanglement-Based Protocols with an Untrusted Source

- Step 1: The untrusted third party, Charlie, initially prepares an entangled source. He sends one mode A
_{1}to Alice through Channel 1 and sends the other mode B_{1}to Bob through Channel 2, where Eve may perform her attack. - Step 2: Alice and Bob perform either a homodyne (switching) (Hom) or a heterodyne (no switching) (Het) measurement on the received modes A
_{2}and B_{2}. Once Alice and Bob have collected a sufficiently large set of correlated data, they proceed with classical data post-processing, namely error reconciliation and privacy amplification. The reconciliation can be performed in one of two ways: either direct reconciliation (DR) [7] or reverse reconciliation (RR) [8].

_{DR}for direct reconciliation and K

_{RR}for reverse reconciliation are given by [41]:

_{E|x}and ρ

_{E|y}are the corresponding state of Eve’s ancillas and ${\rho}_{E}={\displaystyle {\sum}_{x}p(x){\rho}_{E|x}}$ and ${\rho}_{E}={\displaystyle {\sum}_{y}p(y){\rho}_{E|y}}$ are Eve’s average states for DR and RR, respectively. Unless both Alice and Bob performed heterodyne measurements, they first apply a sifting process, where they compare the chosen measurement quadrature $(\widehat{x}\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}\widehat{\rho})$ and only keep the data for which the quadratures match. Here, we use x and y to represent Alice’s and Bob’s measurement results, respectively, for both homodyne and heterodyne measurements.

_{2}|x) for DR and S (E|y) = S (A

_{2}|y) for RR. Thus, χ (A : E) and χ (B : E) become:

_{2}(x+1)−x log

_{2}x, λ

_{1,2}are the symplectic eigenvalues of the covariance matrix ${\gamma}_{{A}_{2}{B}_{2}}$ and λ

_{3}, λ

_{4}are the symplectic eigenvalues of the covariance matrices ${\gamma}_{{B}_{2}|x}$ and ${\gamma}_{{A}_{2}|y}$ [5].

#### 2.2. Entanglement Swapping: Entanglement-Based Protocol with an Untrusted Relay

- Step 1: Alice and Bob both generate an Einstein–Podolsky–Rosen (EPR), states EPR
_{1}and EPR_{2}, respectively, with variances V_{A}and V_{B}and they keep modes A_{2}and B_{2}at their respective sides. Then, they send their other modes A_{1}and B_{1}to the untrusted third party (Charlie) through two different quantum channels with lengths L_{AC}and L_{BC}. - Step 2: Charlie combines the received two modes ${{A}^{\prime}}_{1}$ and ${{B}^{\prime}}_{1}$ onto a beam splitter (50:50), where we label output modes of the beam splitter as C and D. Charlie then measures the x-quadrature of mode C and the p-quadrature of mode D using homodyne detectors and publicly announces the measurement results x
_{C}, p_{D}to Alice and Bob through classical channels. After the measurements of modes C and D, the two initially independent modes A_{2}and B_{2}get entangled if channel noise is not too strong. - Step 3: Bob displaces the mode B
_{2}to B_{3}by the operation $\widehat{D}(\beta )$and gets ${\widehat{\rho}}_{{B}_{3}}=\widehat{D}(\beta ){\widehat{\rho}}_{{B}_{2}}{\widehat{D}}^{\u2020}(\beta )$, where ${\widehat{\rho}}_{B}$ represents the density matrix of mode B, β = g (X_{C}+ iP_{D}), $\widehat{D}(\beta )={e}^{\beta}{}^{{\xe2}^{\u2020}-\beta \ast \xe2}$ (â^{†}and â are the creation and annihilation operators, respectively), and g represents the gain of the displacement. Then Bob measures the mode B_{3}to get the final data {x_{B}, p_{B}} using heterodyne detection. Alice also measures the mode A_{2}to get the final data {x_{A}, p_{A}}, again using heterodyne detection. - Step 4: Once Alice and Bob have collected a sufficiently large set of correlated data, they use an authenticated public channel to do parameter estimation from a randomly-chosen sample of final data from {x
_{A}, p_{A}} and {x_{B}, p_{B}}. Then, Alice and Bob proceed with classical data post-processing to distil a secret key. The reconciliation can also be done in two ways: either DR or RR.

_{2}and B

_{2}by the operators D (α

_{1}) and D (α

_{2}), resulting in ${\rho}_{A}{}_{{}_{3}}=D({\alpha}_{1}){\rho}_{A}{}_{{}_{2}}{D}^{\u2020}({\alpha}_{1})$ and ${\rho}_{B}{}_{{}_{3}}=D({\alpha}_{2}){\rho}_{B}{}_{{}_{2}}{D}^{\u2020}({\alpha}_{2})$, where α

_{1}= −g

_{A}(X

_{C}− iP

_{D})/2, α

_{2}= g

_{B}(X

_{C}+ iP

_{D})/2, and g

_{A}, g

_{B}represents the gain of the displacements at Alice’s and Bob’s side, respectively.

## 3. Improvement Using Noiseless Linear Amplifiers

#### 3.1. Noiseless Linear Amplifier

_{n}is the n × n identity matrix, and σ

_{z}= diag (1, −1).

_{n})

^{−1}Thus, we can find:

_{jk,lm}= ρ

_{jk,lm}/ρ

_{00,00}[49]. Then, the matrix Γ after the two NLAs becomes:

_{1}and g

_{2}are the gains of the NLAs at Alice’s and Bob’s sides (g

_{1}= 1 or g

_{2}= 1 means there is no NLA). Thus, the covariance matrix γ′ after the NLAs can be obtained by:

_{total}= P

_{A}P

_{B|A}, where P

_{A}is the success probability of Alice’s NLA and P

_{B|A}is the success probability of Bob’s NLA given that Alice’s amplification succeeded. Furthermore, considering the trade-off between the fidelity and the success probability of an NLA, a good estimate of the maximal expected success probability for one NLA is given by [50]:

#### 3.2. Entanglement-Based Protocol with an Untrusted Source

_{1}, T

_{2}, and excess noise ε

_{1}, ε

_{2}followed by two gain efficiencies g

_{1}, g

_{2}, is equal to the covariance matrix ${\gamma}_{AB}(\varsigma ,{\eta}_{1},{\epsilon}_{1}^{g},{g}_{1}=1,{\eta}_{2},{\epsilon}_{2}^{g},{g}_{2}=1)$ of an equivalent system with an EPR parameter ς, sent through two channels with parameters η

_{1}, ${\epsilon}_{1}^{g}$ and η

_{2}, ${\epsilon}_{2}^{g}$, without using NLAs. These parameters are given by:

_{1}, ${\epsilon}_{1}^{g}$, η

_{2}and ${\epsilon}_{2}^{g}$ do not depend on λ. Thus, the first condition is always satisfied if λ is below a limiting value, given by:

_{DR}as a function of distance d under four situations: without NLAs (g

_{1}= 1, g

_{2}= 1), with only an NLA at Alice’s side (g

_{2}= 1), with only an NLA at Bob’s side (g

_{1}= 1) and with two NLAs at both sides. The various parameters are chosen from typical experimental values [6]: we choose V = 1.7, β = 0.948 and ε = 0.002 (where the shot noise variance is normalized to one). The transmittance T = 10

^{−}

^{ad}

^{/10}, where a = 0.2 dB/km is the loss coefficient of the optical fibres and d is the length of the quantum channel. The total success probability of using two NLAs for the CV-QKD protocols with EPR in the middle is ${P}_{\text{total}}=1/\left({g}_{1}^{2{N}_{A}}{g}_{2}^{2{N}_{B|A}}\right)$ where N

_{A}= T (V − 1 + ε) + 1, N

_{B|A}= T (V′ − 1 + ε) + 1. Here, V′ is the variance of the equivalent EPR when Alice’s amplification succeeds, which is given by V′ = (1 + ς

^{2})/(1 − ς

^{2}) provided g

_{2}= 1.

_{1}= g

_{2}= 1.4. The NLAs enhance the maximal transmission of the protocol, in which Alice is using heterodyne detection and Bob is using homodyne detection with DR, from 17.0 km to 31.6 km. Furthermore, we also find that if we only put an NLA at either Alice’s or Bob’s side, the performance of the protocols can also be improved. For instance, placing an NLA at the non-reconciliation side (Alice’s side for RR protocols and Bob’s side for DR protocols) has a greater improvement than placing it at the other side. This is because when adding an NLA only at one side (suppose it is on Alice’s side), according to Equation (12), the covariance matrix after the application of the NLA has the feature that Alice’s equivalent variance is greater than Bob’s variance. If considering Alice’s part as the reconciliation part, it is similar to the one-way CV-QKD protocol with DR; while, if considering Bob’s part as the reconciliation part, it is similar to the one-way CV-QKD protocol with RR. In one-way protocols, the RR protocol usually has a longer transmission distance than the DR protocol. Therefore, in our protocols, placing an NLA at the non-reconciliation side is better than placing it at the reconciliation side. Obviously, the optimal performance of the protocols is achieved by placing two NLAs at each side. However, if we want to reduce the cost and expense and only have one NLA in the deployment, we need to place it at the correct side to have the greatest improvement.

#### 3.3. Entanglement-Based Protocol with an Untrusted Relay

_{A}= V

_{B}= 1.7, β = 0.948, ε = ε

_{1}= ε

_{2}=0.002, ${g}_{A}=\sqrt{({V}^{2}-1)/[2{T}_{1}(V+\epsilon )+2(1-{T}_{1})]}$, ${g}_{B}=\sqrt{({V}^{2}-1)/[2{T}_{2}(V+\epsilon )+2(1-{T}_{2})]}$. Under these simulation parameters, the modified entanglement-based protocol in the symmetric case (the distance from Alice to Charlie L

_{AC}is equal to the distance from Bob to Charlie L

_{BC}) can successfully distribute secret keys under such conditions. Then, using the same method as above, we place an NLA at each side to improve its performance; we find an improvement when we set the two gain efficiencies as g

_{1}= g

_{2}= 1.8. The NLAs enhance the maximal transmission distance of the protocol from 1.6 km to 5.3 km in the symmetric case.

_{AB}will increase to a relatively longer distance. Thus, we study the performance of the asymmetric case where L

_{AC}≠ L

_{BC}. As illustrated in Figure 7b, the total maximal transmission distance increases when L

_{AC}decreases. In the asymmetric case, the performance of the modified CV-QKD protocol is also improved by placing two NLAs, one at each side. The maximal total transmission distance of the modified protocol using two NLAs, with gain efficiencies g

_{1}= g

_{2}= 1.8, is enhanced from 17.5 km to 25.2 km in the most asymmetric case (i.e., L

_{AC}≈ 0 km). Here ‘0 km’ indicates that the transmission distance from Alice to Charlie is very short but not exactly zero. In fact, even when Charlie is at Alice’s side, there still exists a distance between Alice’s laser and the beamsplitter. Therefore, in the numerical simulation although we assume the channel transmittance is T

_{1}= 1, the excess noise ε

_{1}still exists, and is ε

_{1}= 0.002.

## 4. Conclusion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Schematic of the continuous-variable version of quantum key distribution (CV-QKD) protocols with an untrusted source. Both the entangled Gaussian source and the quantum channels are fully controlled by Eve. However, Eve has no access to the apparatuses in Alice’s and Bob’s stations. Alice and Bob can perform either homodyne (Hom) or heterodyne (Het) detection, using either direct or reverse reconciliation.

**Figure 2.**(

**a**) Entanglement-based CV-QKD protocol with an untrusted relay where the displacement operator is placed at Bob’s side. (

**b**) Entanglement-based scheme where the displacement operator is placed at both Alice’s and Bob’s sides.

**Figure 3.**Entanglement-in-the-middle protocol. Equivalent channels and squeezing: an Einstein–Podolsky–Rosen (EPR) state λ sent through two Gaussian channels of transmittance T

_{1}, T

_{2}and excess noise ε

_{1}, ε

_{2}, followed by two successful noiseless linear amplifiers (NLAs), has the same final covariance matrix with a state ς sent through two Gaussian channels of transmittance η

_{1}, η

_{2}and excess noise ${\epsilon}_{1}^{g}$, ${\epsilon}_{2}^{g}$, without two NLAs.

**Figure 4.**Improvement of the CV-QKD protocols with entanglement-in-the-middle. A comparison among the secret key rates for the protocols where (left panel) Alice uses homodyne detection and Bob uses homodyne detection and DR(equivalent to Alice using homodyne detection and Bob using homodyne detection and RR); and (right panel) Alice uses heterodyne detection and Bob uses homodyne detection and DR (equivalent to Alice using homodyne detection and Bob using heterodyne detection and RR), under the following situations: no NLAs (g

_{1}= 1, g

_{2}= 1), using an NLA at Alice’s side (g

_{2}= 1), using an NLA at Bob’s side (g

_{1}= 1) and using two NLAs at both sides. Here, we use the realistic parameters: V = 1.7, β = 0.948, ε = 0.002 and ${P}_{\text{total}}=1/\left({g}_{1}^{2{N}_{A}}{g}_{2}^{2{N}_{B|A}}\right)$.

**Figure 5.**Improvement of the CV-QKD protocols with entanglement-in-the-middle. A comparison among the secret key rates for the protocols where (left panel) Alice uses homodyne detection and Bob uses heterodyne detection and DR (equivalent to Alice using heterodyne detection and Bob using homodyne detection and RR); and (right panel) Alice uses heterodyne detection and Bob uses heterodyne detection and DR (equivalent to Alice using heterodyne detection and Bob using heterodyne detection and RR), under the following situations: no NLAs (g

_{1}= 1, g

_{2}= 1), using an NLA at Alice’s side (g

_{2}= 1), using an NLA at Bob’s side (g

_{1}= 1) and using two NLAs at both sides. Here, we use the realistic parameters: V = 1.7, β = 0.948, ε = 0.002 and ${P}_{\text{total}}=1/\left({g}_{1}^{2{N}_{A}}{g}_{2}^{2{N}_{B|A}}\right)$.

**Figure 6.**Entanglement-based scheme of the modified CV-QKD protocol with an untrusted relay, where a displacement operator D is placed at both Alice’s and Bob’s sides and the two NLAs are placed before the measurement devices.

**Figure 7.**Improvement of the modified CV-QKD protocol with an untrusted relay in (

**a**) the symmetric case (i.e., L

_{AC}= L

_{BC}) and (

**b**) the asymmetric case (i.e., L

_{AC}≠ L

_{BC}). A comparison among the secret key rates in DR under the following situations: no NLAs (g

_{1}= 1, g

_{2}= 1), using an NLA at Alice’s side (g

_{2}= 1), using an NLA at Bob’s side (g

_{1}= 1) and using two NLAs one at each side. Here, we use the realistic parameters: V

_{A}= V

_{B}= 1.7, β = 0.948, ε = 0.002 and ${P}_{\text{total}}=1/\left({g}_{1}^{2{N}_{A}}{g}_{2}^{2{N}_{B|A}}\right)$.

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## Share and Cite

**MDPI and ACS Style**

Zhang, Y.; Li, Z.; Weedbrook, C.; Marshall, K.; Pirandola, S.; Yu, S.; Guo, H.
Noiseless Linear Amplifiers in Entanglement-Based Continuous-Variable Quantum Key Distribution. *Entropy* **2015**, *17*, 4547-4562.
https://doi.org/10.3390/e17074547

**AMA Style**

Zhang Y, Li Z, Weedbrook C, Marshall K, Pirandola S, Yu S, Guo H.
Noiseless Linear Amplifiers in Entanglement-Based Continuous-Variable Quantum Key Distribution. *Entropy*. 2015; 17(7):4547-4562.
https://doi.org/10.3390/e17074547

**Chicago/Turabian Style**

Zhang, Yichen, Zhengyu Li, Christian Weedbrook, Kevin Marshall, Stefano Pirandola, Song Yu, and Hong Guo.
2015. "Noiseless Linear Amplifiers in Entanglement-Based Continuous-Variable Quantum Key Distribution" *Entropy* 17, no. 7: 4547-4562.
https://doi.org/10.3390/e17074547