Generation of Three-Mode Entanglement Based on Parametric Amplifiers Using Quantum Entanglement Swapping

Abstract

Quantum entanglement swapping is one of the most promising quantum techniques to create or manipulate large-scale multi-mode entanglement between two distant quantum entangled systems. In this work, a scheme for the generation of three-mode entanglement based on parametric amplifiers using quantum entanglement swapping has been proposed. The newly generated three-mode entanglement is always present in the whole power gain region from parametric amplifiers and its dependence on transmission loss and feedforward gain is also investigated. In addition, the effects of power gain, transmission loss, and feedforward gain on the two-mode entanglement of the three pairs in the newly generated three-mode entanglement have also been analysed in detail. The results presented here may find some practical applications in quantum secure communication.

1. Introduction

Quantum entanglement swapping [1,2,3,4,5,6,7,8], which makes two independent quantum entangled systems become entangled without any direct interaction, is an important technique in constructing large-scale quantum networks. It can be expressed in the simplest two entangled pairs which are generated at two distant locations $\alpha$ and $\beta$ respectively, i.e., (${\alpha }_1$ and ${\alpha }_2$) and (${\beta }_1$ and ${\beta }_2$). Then, bell-state measurement is applied to the two modes ${\alpha }_1$ and ${\beta }_1$, which will project ${\alpha }_2$ and ${\beta }_2$ in an entangled state, although they may have never interacted with each other before. The applications for quantum entanglement swapping have been expanded to the creation and manipulation of multi-mode entanglement from that of two-mode entanglement [9]. For example, quantum entanglement swapping between two independent Greenberger–Horne–Zeilinger (GHZ) entangled states in Ref. [10], in which three-mode GHZ entangled states and bipartite entangled states before quantum entanglement swapping were prepared by mixing several single-mode squeezed states on linear beam splitters, the emphasis of which was on the generation of single-mode squeezed states before implementing quantum entanglement swapping; quantum entanglement swapping between three pairs of entangled photons for the generation of a GHZ state in Ref. [11]; and quantum entanglement swapping between discrete domain and continuous variable (CV) domain [12]. Due to its power for entangling distant modes, quantum entanglement swapping has also found an important role in quantum secure communication [13].

Four-wave mixing (FWM) as one type of parametric amplifier, has proven to be an efficient technique to generate two-mode entanglement in the CV domain [14,15,16,17,18,19,20,21,22,23,24]. Furthermore, genuine tripartite entanglement has been theoretically proved to be present in the whole power gain region in the cascaded FWM processes using positivity under partial transposition (PPT) criterion [25]. Therefore, a scheme for the generation of three-mode entanglement using quantum entanglement swapping between tripartite entanglement in the cascaded FWM processes and bipartite entanglement in a single FWM process has been proposed, the possibility for the generation of three-mode entanglement and its dependence on transmission loss and feedforward gain are discussed. As compared to Ref. [10], in our present scheme the generation of a CV three-mode entangled state instead of a GHZ state and bipartite entangled state relies on nonlinear interaction or mixing between several vacuum state inputs and it lays stress on nonlinear interaction by means of parametric amplifiers. Besides, the present scheme can be easily extended to construct large-scale quantum networks, e.g., four-mode entanglement generation based on two tripartite entanglement [25], five-mode entanglement generation based on tripartite entanglement and quadripartite entanglement [26], and six-mode entanglement generation based on two quadripartite entanglement, etc.

This work is organized as follows: In Section 2, we describe the scheme for the generation of three-mode entanglement using quantum entanglement swapping. In Section 3, the possibility of the generation of three-mode entanglement and its dependence on system parameters, i.e., transmission loss and feedforward gain, are discussed. In Section 4, the effect of system parameters on the two-mode entanglement of the three pairs from the newly generated three-mode entanglement is also discussed. In Section 5, we give a brief summary of this work.

2. The Scheme for the Generation of Three-Mode Entanglement

The schematic diagram of the generation of three-mode entanglement is depicted in Figure 1. Two independent quantum entangled systems A and B, are merged into three-mode entangled systems (${\widehat{B}}_1$, ${\widehat{A}}_2$, and ${\widehat{A}}_3$). In order to construct three-mode entanglement between two independent quantum entangled systems, optical mode ${\widehat{A}}_1$ is sent to bipartite entangled system B through a quantum channel with transmission loss $\eta$ in which $\eta$ is used to simulate the unavoidable losses in real quantum channels which will deteriorate the performance of quantum entanglement swapping. A joint measurement on the two modes ${\widehat{A}}_1$ and ${\widehat{B}}_2$ is implemented and then the measured results are fed forward with the gain $\beta$ to the optical mode ${\widehat{B}}_1$. The remaining optical modes of the two independent quantum entangled systems, i.e., ${\widehat{B}}_1$, ${\widehat{A}}_2$, and ${\widehat{A}}_3$, are projected and entangled. The three-mode entanglement between ${\widehat{B}}_1$, ${\widehat{A}}_2$, and ${\widehat{A}}_3$ can be analysed in the detailed setup in Figure 2. As depicted in Figure 2, the input–output relationship of the cascaded FWM processes can be expressed as

$$\begin{array}{ccc}\hfill {\widehat{A}}_1& =& \sqrt{G_1\eta }{\widehat{a}}_{v1}+\sqrt{g_1\eta }{\widehat{a}}_0^{†}+\sqrt{1-\eta }{\widehat{a}}_{v4},\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill {\widehat{A}}_2& =& \sqrt{G_1{G}_2}{\widehat{a}}_0+\sqrt{G_2{g}_1}{\widehat{a}}_{v1}^{†}+\sqrt{g_2}{\widehat{a}}_{v2}^{†},\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill {\widehat{A}}_3& =& \sqrt{G_2}{\widehat{a}}_{v2}+\sqrt{G_1{g}_2}{\widehat{a}}_0^{†}+\sqrt{g_1{g}_2}{\widehat{a}}_{v1},\hfill \end{array}$$

(3)

where $\eta$ is transmission loss, Gj (j = 1, 2) is the power gain in the cascaded FWM processes, and Gj$-g$j = 1. Meanwhile, the input–output relationship of the single FWM process can be expressed as

$$\begin{array}{ccc}\hfill {\widehat{B}}_1& =& \sqrt{G_3}{\widehat{a}}_{v3}+\sqrt{g_3}{\widehat{a}}_1^{†},\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill {\widehat{B}}_2& =& \sqrt{G_3}{\widehat{a}}_1+\sqrt{g_3}{\widehat{a}}_{v3}^{†},\hfill \end{array}$$

(5)

where G3 is the power gain of the single FWM process and G3$-g$3 = 1.

Figure 1. The scheme of the generation of three-mode entanglement. ${\widehat{A}}_1$, ${\widehat{A}}_2$, and ${\widehat{A}}_3$: tripartite entanglement in the cascaded FWM processes A; ${\widehat{B}}_1$ and ${\widehat{B}}_2$: bipartite entanglement in a single FWM process B; $\eta$: transmission loss; $\beta$: feedforward gain.

Figure 2. Detailed setup for the generation of three-mode entanglement using quantum entanglement swapping. ${\widehat{a}}_0$ and ${\widehat{a}}_1$: coherent input signals; ${\widehat{a}}_{\nu 1}$, ${\widehat{a}}_{\nu 2}$, ${\widehat{a}}_{\nu 3}$, and ${\widehat{a}}_{\nu 4}$: vacuum inputs; ${\widehat{A}}_1$, ${\widehat{A}}_2$, ${\widehat{A}}_3$: triple output modes in the cascaded FWM system; ${\widehat{B}}_1$ and ${\widehat{B}}_2$: two output modes in the single FWM system; ${\widehat{C}}_1$: mode ${\widehat{B}}_1$ after quantum entanglement swapping; $G_1$, $G_2$, and $G_3$: the power gain values of FWM${}_1$, FWM${}_2$, and FWM${}_3$, respectively; $\eta$: transmission loss; 50:50: beam splitter; 1:99: highly reflecting mirrors; $\beta$: feedforward gain; HD: homodyne detector; EOM: electro-optic modulator.

Equations (1)–(5) can also be expressed in terms of the quadrature operators as follows

$$\begin{array}{ccc}\hfill X_{{\widehat{A}}_1}& =& {\widehat{A}}_1+{\widehat{A}}_1^{†}\hfill \\ & =& \sqrt{G_1\eta }{X}_{v1}+\sqrt{g_1\eta }{X}_0+\sqrt{1-\eta }{X}_{v4},\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill X_{{\widehat{A}}_2}& =& {\widehat{A}}_2+{\widehat{A}}_2^{†}\hfill \\ & =& \sqrt{G_1{G}_2}{X}_0+\sqrt{G_2{g}_1}{X}_{v1}+\sqrt{g_2}{X}_{v2},\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill X_{{\widehat{A}}_3}& =& {\widehat{A}}_3+{\widehat{A}}_3^{†}\hfill \\ & =& \sqrt{G_2}{X}_{v2}+\sqrt{G_1{g}_2}{X}_0+\sqrt{g_1{g}_2}{X}_{v1},\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill Y_{{\widehat{A}}_1}& =& -i({\widehat{A}}_1-{\widehat{A}}_1^{†})\hfill \\ & =& \sqrt{G_1\eta }{Y}_{v1}-\sqrt{g_1\eta }{Y}_0+\sqrt{1-\eta }{Y}_{v4},\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill Y_{{\widehat{A}}_2}& =& -i({\widehat{A}}_2-{\widehat{A}}_2^{†})\hfill \\ & =& \sqrt{G_1{G}_2}{Y}_0-\sqrt{G_2{g}_1}{Y}_{v1}-\sqrt{g_2}{Y}_{v2},\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill Y_{{\widehat{A}}_3}& =& -i({\widehat{A}}_3-{\widehat{A}}_3^{†})\hfill \\ & =& \sqrt{G_2}{Y}_{v2}-\sqrt{G_1{g}_2}{Y}_0+\sqrt{g_1{g}_2}{Y}_{v1},\hfill \end{array}$$

(11)

and

$$\begin{array}{ccc}\hfill X_{{\widehat{B}}_1}& =& {\widehat{B}}_1+{\widehat{B}}_1^{†}=\sqrt{G_3}{X}_{v3}+\sqrt{g_3}{X}_1,\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill X_{{\widehat{B}}_2}& =& {\widehat{B}}_2+{\widehat{B}}_2^{†}=\sqrt{G_3}{X}_1+\sqrt{g_3}{X}_{v3},\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill Y_{{\widehat{B}}_1}& =& -i({\widehat{B}}_1-{\widehat{B}}_1^{†})=\sqrt{G_3}{Y}_{v3}-\sqrt{g_3}{Y}_1,\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill Y_{{\widehat{B}}_2}& =& -i({\widehat{B}}_2-{\widehat{B}}_2^{†})=\sqrt{G_3}{Y}_1-\sqrt{g_3}{Y}_{v3},\hfill \end{array}$$

(15)

then, the mode ${\widehat{A}}_1$ from the tripartite entangled system A is transmitted to bipartite entangled system B and mixed with ${\widehat{B}}_2$ on a 50:50 beam splitter. The output modes are measured by two homodyne detectors (HDs), and the measured photocurrents for the quadratures $X_{\mu }$ and $Y_{\nu }$ can be written as

$$\begin{array}{ccc}\hfill X_{\mu }& =& \frac{1}{\sqrt{2}}(X_{{\widehat{A}}_1}-X_{{\widehat{B}}_2}),\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill Y_{\nu }& =& \frac{1}{\sqrt{2}}(Y_{{\widehat{A}}_1}+Y_{{\widehat{B}}_2}),\hfill \end{array}$$

(17)

respectively. The measured results of $\sqrt{2}\beta X_{\mu }$ and $\sqrt{2}\beta Y_{\nu }$ are fed forward to the mode ${\widehat{B}}_1$ from bipartite entangled system. Thus, Equations (12) and (14) can be changed into

$$\begin{array}{ccc}\hfill X_{{\widehat{C}}_1}& =& X_{{\widehat{B}}_1}+\beta (X_{{\widehat{A}}_1}-X_{{\widehat{B}}_2})\hfill \\ & =& \beta X_{{\widehat{A}}_1}+(X_{{\widehat{B}}_1}-\beta X_{{\widehat{B}}_2}),\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill Y_{{\widehat{C}}_1}& =& Y_{{\widehat{B}}_1}+\beta (Y_{{\widehat{A}}_1}+Y_{{\widehat{B}}_2})\hfill \\ & =& \beta Y_{{\widehat{A}}_1}+(Y_{{\widehat{B}}_1}+\beta Y_{{\widehat{B}}_2}),\hfill \end{array}$$

(19)

as we can see from Equations (18) and (19), $X_{{\widehat{B}}_1}-\beta X_{{\widehat{B}}_2}$ and $Y_{{\widehat{B}}_1}+\beta Y_{{\widehat{B}}_2}$ represent Einstein–Podolsky–Rosen (EPR) correlations in a single FWM process, thus the value of $\beta$ is better to equal to 1.

The above quantum entanglement swapping process utilizes the two modes ${\widehat{A}}_1$ and ${\widehat{B}}_2$ to entangle the remaining modes ${\widehat{A}}_2$, ${\widehat{A}}_3$, and ${\widehat{C}}_1$. The newly generated three-mode entanglement between ${\widehat{A}}_2$, ${\widehat{A}}_3$, and ${\widehat{C}}_1$ can be discussed using PPT criterion. Thus, the covariance matrix (CM) for the three modes should be obtained firstly. It can be obtained from Equations (7), (8), (10), (11), (18) and (19)

$$\begin{array}{ccc}\hfill ⟨X_{{\widehat{C}}_1}^2⟩& =& ⟨Y_{{\widehat{C}}_1}^2⟩=\eta {\beta }^2(2G_1-1)+{\beta }^2(1-\eta )\hfill \\ & & +{(\sqrt{G_3}-\beta \sqrt{G_3-1})}^2\hfill \\ & & +{(\sqrt{G_3-1}-\beta \sqrt{G_3})}^2,\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill ⟨X_{{\widehat{A}}_2}^2⟩& =& ⟨Y_{{\widehat{A}}_2}^2⟩=2G_1{G}_2-1,\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill ⟨X_{{\widehat{A}}_3}^2⟩& =& ⟨Y_{{\widehat{A}}_3}^2⟩=2G_1{G}_2-2G_1+1,\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill ⟨X_{{\widehat{C}}_1}{X}_{{\widehat{A}}_2}⟩& =& -⟨Y_{{\widehat{C}}_1}{Y}_{{\widehat{A}}_2}⟩=2\beta \sqrt{G_1{G}_2(G_1-1)\eta },\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill ⟨X_{{\widehat{C}}_1}{X}_{{\widehat{A}}_3}⟩& =& ⟨Y_{{\widehat{C}}_1}{Y}_{{\widehat{A}}_3}⟩\hfill \\ & =& 2\beta \sqrt{G_1(G_1-1)(G_2-1)\eta },\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill ⟨X_{{\widehat{A}}_2}{X}_{{\widehat{A}}_3}⟩& =& -⟨Y_{{\widehat{A}}_2}{Y}_{{\widehat{A}}_3}⟩=2G_1\sqrt{G_2(G_2-1)},\hfill \end{array}$$

(25)

where $⟨X_m^2⟩$ and $⟨X_m{X}_n⟩$ denote the amplitude quadrature variance and the covariance, respectively. For the case where $m=n$, the covariance, denoted as $⟨X_m{X}_n⟩$, reduces to the usual variance, $⟨X_m^2⟩$. The variances and covariances in Equations (20)–(25) contain all the information about the newly generated three-mode entanglement, and they can be used to analyse the quantum entanglement performance of three-mode and two-mode entanglement.

3. Three-Mode Entanglement

In this section, the newly generated three-mode entanglement will be analysed using PPT criterion. PPT criterion is a necessary and sufficient criterion for all 1 × N decompositions of Gaussian states, where 1 + N is the total number of the entangled modes [27,28,29,30,31,32]. For three-mode entanglement conditions, the three possible 1 × 2 partitions have to be tested. All the partitions of three-mode state are inseparable when the smallest symplectic eigenvalues for each of the three partially transposed (PT) states are all smaller than 1, i.e., genuine three-mode entanglement is present. Firstly, in Ref. [25], tripartite entanglement, i.e., ${\widehat{A}}_1$, ${\widehat{A}}_2$, and ${\widehat{A}}_3$ without transmission loss $\eta$, has been proven to be always present in the whole power gain region ($G_1>1$ and $G_2>1$) in the cascaded FWM processes. Secondly, bipartite entanglement between ${\widehat{B}}_1$ and ${\widehat{B}}_2$ can be quantified by the smaller symplectic eigenvalue of PT CM${}_{{\widehat{B}}_1{\widehat{B}}_2}$

$$\begin{array}{c}\hfill E_{{\widehat{B}}_1{\widehat{B}}_2}={(\sqrt{G_3}-\sqrt{G_3-1})}^2,\end{array}$$

(26)

It can be easily proven that the value of $E_{{\widehat{B}}_1{\widehat{B}}_2}$ is always smaller than 1 only if $G_3>1$ in the single FWM process. Finally, to demonstrate the presence of the newly generated three-mode entanglement (${\widehat{C}}_1$, ${\widehat{A}}_2$, and ${\widehat{A}}_3$), the power gain values $G_1$ and $G_2$ in the cascaded FWM processes are fixed and set equal to 2, the transmission loss $\eta$ and feedforward gain $\beta$ are both set equal to 1. Thus, the newly generated three-mode entanglement only depends on power gain G${}_3$. The results are depicted in Figure 3, the traces A ($E_{{\widehat{C}}_1-{\widehat{A}}_2{\widehat{A}}_3}$), B ($E_{{\widehat{A}}_3-{\widehat{C}}_1{\widehat{A}}_2}$), and C ($E_{{\widehat{A}}_2-{\widehat{C}}_1{\widehat{A}}_3}$) are the smallest symplectic eigenvalues when PT operation is applied to the modes ${\widehat{C}}_1$, ${\widehat{A}}_3$, and ${\widehat{A}}_2$, respectively. Their detailed expressions are not shown here due to their complexity. As we can see from Figure 3, the three smallest symplectic eigenvalues decrease with the increasing power gain $G_3$ and are always smaller than 1 only if $G_3>1$. It means that genuine three-mode entanglement (${\widehat{C}}_1$, ${\widehat{A}}_2$, and ${\widehat{A}}_3$) is always present in the whole power gain region, i.e., quantum entanglement swapping between tripartite entanglement in the cascaded FWM processes and bipartite entanglement in the single FWM process can be realized completely.

Figure 3. The dependence of smallest symplectic eigenvalues $E_{{\widehat{C}}_1-{\widehat{A}}_2{\widehat{A}}_3}$ (A), $E_{{\widehat{A}}_3-{\widehat{C}}_1{\widehat{A}}_2}$ (B), and $E_{{\widehat{A}}_2-{\widehat{C}}_1{\widehat{A}}_3}$ (C) on power gain G${}_3$ under the conditions of $G_1$ = $G_2$ = 2, $\eta =1$, $\beta =1$.

After the possibility of the generation of three-mode entanglement is demonstrated, the dependence of three-mode entanglement on transmission loss $\eta$ under the conditions of $G_1$ = $G_2$ = $G_3$ = 2 and $\beta =1$ will be discussed and the results are depicted in Figure 4. As depicted in Figure 4, the values of $E_{{\widehat{A}}_3-{\widehat{C}}_1{\widehat{A}}_2}$ and $E_{{\widehat{A}}_2-{\widehat{C}}_1{\widehat{A}}_3}$ slightly decrease with the increasing of transmission loss $\eta$ and are always smaller than 1, while the value of $E_{{\widehat{C}}_1-{\widehat{A}}_2{\widehat{A}}_3}$ becomes smaller than 1 only when the value of $\eta$ exceeds one critical value. This also means that the newly generated three-mode entanglement after quantum entanglement swapping evolves from a partial inseparability state to a full inseparability state, and it is also not a GHZ state due to the mathematical formulas in Equations (2) and (3), different from those of a GHZ state. Different from the fragility of $E_{{\widehat{C}}_1-{\widehat{A}}_2{\widehat{A}}_3}$, the immunity of $E_{{\widehat{A}}_3-{\widehat{C}}_1{\widehat{A}}_2}$ and $E_{{\widehat{A}}_2-{\widehat{C}}_1{\widehat{A}}_3}$ to $\eta$ is because bipartite entanglement between the two modes ${\widehat{A}}_2$ and ${\widehat{A}}_3$ from one FWM process is always present. However, genuine three-mode entanglement is present only when the value of $\eta$ exceeds one critical value.

Figure 4. The dependence of smallest symplectic eigenvalues $E_{{\widehat{C}}_1-{\widehat{A}}_2{\widehat{A}}_3}$ (A), $E_{{\widehat{A}}_3-{\widehat{C}}_1{\widehat{A}}_2}$ (B), and $E_{{\widehat{A}}_2-{\widehat{C}}_1{\widehat{A}}_3}$ (C) on transmission loss $\eta$ under the condition of $G_1$ = $G_2$ = $G_3$ = 2 and $\beta =1$.

Similarly, the dependence of three-mode entanglement on feedforward gain $\beta$ under the conditions of $G_1$ = $G_2$ = $G_3$ = 2 and $\eta =1$ will be analysed and the results are depicted in Figure 5. As depicted in Figure 5, the minimum values of $E_{{\widehat{C}}_1-{\widehat{A}}_2{\widehat{A}}_3}$ (A), $E_{{\widehat{A}}_3-{\widehat{C}}_1{\widehat{A}}_2}$ (B), and $E_{{\widehat{A}}_2-{\widehat{C}}_1{\widehat{A}}_3}$ (C) can be obtained under the condition of $\beta \simeq 1$, which is in agreement with Equations (18) and (19).

Figure 5. The dependence of smallest symplectic eigenvalues $E_{{\widehat{C}}_1-{\widehat{A}}_2{\widehat{A}}_3}$ (A), $E_{{\widehat{A}}_3-{\widehat{C}}_1{\widehat{A}}_2}$ (B), and $E_{{\widehat{A}}_2-{\widehat{C}}_1{\widehat{A}}_3}$ (C) on feedforward gain $\beta$ under the conditions of $G_1$ = $G_2$ = $G_3$ = 2 and $\eta =1$.

4. Two-Mode Entanglement

In this section, two-mode entanglement for the three pairs from the newly generated three-mode entanglement will be discussed. Firstly, the effect of power gain $G_3$ on two-mode entanglement under the conditions of $G_1=G_2=2$ and $\eta =\beta =1$ is depicted in Figure 6. As depicted in Figure 6, two-mode entanglement between ${\widehat{C}}_1$ and ${\widehat{A}}_3$ (trace A) is always absent regardless of the value of $G_3$; this is also in agreement with the conclusions about the two modes ${\widehat{A}}_1$ and ${\widehat{A}}_3$ in Ref. [25]. Two-mode entanglement between ${\widehat{C}}_1$ and ${\widehat{A}}_2$ (trace B) is present only when $G_3$ exceeds one critical value, this is because quantum entanglement swapping between the two modes ${\widehat{A}}_1$ and ${\widehat{C}}_1$ requires the support from EPR entanglement in the single FWM process. It is very easy to understand the independence of two-mode entanglement between ${\widehat{A}}_2$ and ${\widehat{A}}_3$ (trace C) on $G_3$. This is because ${\widehat{A}}_2$ and ${\widehat{A}}_3$ are only from the cascaded FWM processes.

Figure 6. The dependence of smallest symplectic eigenvalues $E_{{\widehat{C}}_1-{\widehat{A}}_3}$ (A), $E_{{\widehat{C}}_1-{\widehat{A}}_2}$ (B), and $E_{{\widehat{A}}_2-{\widehat{A}}_3}$ (C) on power gain G${}_3$ under the condition of $G_1=G_2=2$ and $\eta =\beta =1$.

Secondly, the effect of feedforward gain $\beta$ on two-mode entanglement under the conditions of $G_1=G_2=G_3=2$ and $G_1=G_2=G_3=20$ is depicted in Figure 7a and Figure 7b, respectively. As depicted in Figure 7, under the condition of a larger power gain, the value of $\beta$ in which the minimum values of $E_{{\widehat{C}}_1-{\widehat{A}}_3}$ (A), $E_{{\widehat{C}}_1-{\widehat{A}}_2}$ (B), and $E_{{\widehat{A}}_2-{\widehat{A}}_3}$ (C) can be obtained tends to 1.

Figure 7. The dependence of smallest symplectic eigenvalues $E_{{\widehat{C}}_1-{\widehat{A}}_3}$ (A), $E_{{\widehat{C}}_1-{\widehat{A}}_2}$ (B), and $E_{{\widehat{A}}_2-{\widehat{A}}_3}$ (C) on feedforward gain $\beta$ under the conditions of $\eta =1$ and $G_1=G_2=G_3=2$ in (a), $G_1=G_2=G_3=20$ in (b).

Thirdly, the effect of transmission loss $\eta$ on two-mode entanglement under the condition of $G_1=G_2=G_3=2$ and $\beta =1$ is depicted in Figure 8. As depicted in Figure 8, the value of $E_{{\widehat{C}}_1-{\widehat{A}}_3}$ (A) increases with increasing $\eta$, the value of $E_{{\widehat{C}}_1-{\widehat{A}}_2}$ (B) decreases with increasing $\eta$, and the value of $E_{{\widehat{A}}_2-{\widehat{A}}_3}$ (C) remains unchanged with increasing $\eta$.

Figure 8. The dependence of smallest symplectic eigenvalues $E_{{\widehat{C}}_1-{\widehat{A}}_3}$ (A), $E_{{\widehat{C}}_1-{\widehat{A}}_2}$ (B), and $E_{{\widehat{A}}_2-{\widehat{A}}_3}$ (C) on transmission loss $\eta$ under the conditions of $G_1=G_2=G_3=2$ and $\beta =1$.

5. Conclusions

We propose a scheme for the generation of three-mode entanglement using quantum entanglement swapping between tripartite entanglement in cascaded FWM processes and bipartite entanglement in single FWM processes. The possibility for the generation of three-mode entanglement is demonstrated, and its dependence on transmission loss and feedforward gain are discussed. In addition, two-mode entanglement of the three pairs in the newly generated three-mode entanglement has also been studied in detail. The results presented here may find their practical applications in constructing larger-scale quantum networks.