⇦Chinese Physics Letters, 2017, Vol. 34, No. 7, Article code 070301 General Single-Mode Gaussian Operation with Two-Mode Entangled State * Shu-Hong Hao(郝树宏), Xian-Shan Huang(黄仙山), Dong Wang(王东)** Affiliations School of Mathematics and Physics, Anhui University of Technology, Maanshan 243000 Received 22 February 2017 *Supported by the National Natural Science Foundation of China under Grant Nos 61205115, 11474003 and 61675006, the Natural Science Foundation of Anhui Province under Grant Nos 1608085MF133 and 1408085MA19, the Foundation for the Young Talent of Anhui Province under Grant No gxyqZD2016065, and the Youth Foundation of Anhui University of Technology under Grant Nos RD16100249.
**Corresponding author. Email: wangdong@ahut.edu.cn Citation Text: Hao S H, Huang X S and Wang D 2017 Chin. Phys. Lett. 34 070301 Abstract Realizing the logic operations with small-scale states is pursued to improve the utilization of quantum resources and to simplify the experimental setup. We propose a scheme to realize a general single-mode Gaussian operation with a two-mode entangled state by utilizing only one nondegenerate optical parametric amplifier and by adjusting four angle parameters. The fidelity of the output mode can be optimized by changing one of the angle parameters. This scheme would be utilized as a basic efficient element in the future large-scale quantum computation. DOI:10.1088/0256-307X/34/7/070301 PACS:03.67.Lx, 42.50.Dv © 2017 Chinese Physics Society Article Text Quantum computation (QC) shows much superiority in solving some computational problems and simulating quantum systems.[1-5] It offers new insight into the quantum information. The one-way QC model is one of the practical protocols,[6,7] in which the logical operations are achieved through measurements and classical feedforward of the measured results with a cluster entangled state. Considering the collapse effect of the quantum measurement, the one-way QC is irreversible.[6,7] The continuous variable (CV) cluster states can be prepared unconditionally,[8,9] which leads to a deterministically performed QC process.[7,10] In CV one-way QC, Gaussian transformations transform one Gaussian state to another one. These transformations correspond to Hamiltonians which are quadratic in the quadrature amplitude and phase operators for the quantized optical modes. Some Gaussian operations were experimentally realized using a four-mode linear cluster state, for example, the controlled-X gate,[11] the squeezing operation[12] and the controlled-phase gate.[13] Moreover, the rotation operation,[14] the squeezing operation and the Fourier transform[15] have been efficiently realized based on an Einstein–Podolsky–Rosen (EPR) entangled state. As the QC research requirement increases, a complex QC process will compose of multiple logic gates, each of which may consume a large amount of entangled state resources.[16] Considering that the preparation of spatial separated large-scale quantum entanglement states needs amounts of squeezing states and beam splitters,[17,18] the more efficient scheme to realize various logic gates arouses the research interest. If the logic gate sequences consist of a number of single-mode Gaussian operations, they can be simplified to a new single-mode Gaussian operation. Because each sub-operation will consume some quantum resources, it is worth realizing an equivalence operation instead of logic gate sequences. Therefore, the realization of a general single-mode Gaussian operation is necessary. It has been demonstrated that a four-mode linear cluster state is enough to implement a general single-mode Gaussian transformation.[12,19] The scheme uses four balanced homodyne detections (HD) and adjusts the phase difference in each HD. In this Letter, a scheme of a general single-mode Gaussian operation is proposed, which uses only one two-mode entangled state. To achieve this efficient purpose, we introduce four independent and adjustable angle parameters in the step of quantum measurement. Two angle parameters are also from the HDs. The other two added before the HDs are the key parts to improve the resource utilization. Operator $\hat {a}$ is always selected to describe a quantum optical mode, which satisfies the usual bosonic commutator $[\hat {a},\hat {a}^† ]=1$. When the convention $\hbar=1/2$ is chosen, its amplitude and phase quadratures can be expressed as $\hat {x}=(\hat {a}+\hat {a}^†)/2$ and $\hat {p}=(\hat {a}-\hat {a}^†)/2i$, respectively. Gaussian transformations play the roles of the Clifford gates for CV.[19] In the following we restrict ourselves to unitary Gaussian transformations on a single mode. Thus the single-mode Gaussian transformation can be written as $U=e^{i({\alpha \hat {x}^2+\beta \hat {p}^2+\gamma \hat {x}\hat {p}})}$, where $\alpha$, $\beta$ and $\gamma$ are arbitrary real values. After this Gaussian transformation, $\hat {x}$ and $\hat {p}$ become $U^† (\hat {x},\hat {p})^TU=L(\hat {x},\hat {p})^T$, where the 2$\times$2 matrix $L$ is a faithful representation of the symplectic group $Sp(2,R)$ with 3 degrees of freedom, and the criterion ${\rm Det}(L)=1$ is met.[19] Generally, the transformation matrix $L$ can be written as $L=\left(\begin{matrix} a & {-b} \\ d & c \\ \end{matrix}\right)$, where $ac+bd=1$. Considering the commutator relationship $[\hat {x},\hat {p}]=i/2$, when $\beta \alpha -\gamma ^2/4>0$ and taking $C=\sqrt {|{\beta \alpha -\gamma ^2/4}|}$, the transformation matrix $L$ should be expressed as $$\begin{alignat}{1} \!\!\!\!\!\!\left(\begin{matrix} a & {-b} \\ d & c \\ \end{matrix}\right)=\left(\begin{matrix} {\cos C-\frac{\sin C}{2C}\gamma} & {-\beta \frac{\sin C}{C}} \\ {\alpha \frac{\sin C}{C}} & {\cos C+\frac{\sin C}{2C}\gamma} \\ \end{matrix}\right).~~ \tag {1} \end{alignat} $$ Similarly, when $\beta \alpha -\gamma ^2/4\leqslant 0$, we can obtain $L$ just by changing the term $\cos C$ to $\cosh C$ and changing $\sin C$ to $\sinh C$.
cpl-34-7-070301-fig1.png
Fig. 1. (Color online) Schematic diagram of a single-mode quantum logic operation with a two-mode entangled state. E1 (E2): the submode of the two-mode entangled state; LO: the local oscillator for the homodyne detection (HD); $D(x)$ and $D(p)$: the amplitude and phase displacement operators; NOPA: nondegenerate optical parametric amplifier; G: feedforward circuit; HWP: half-wave plate; and PBS: polarization beam splitter. The parameters $\theta _1$, $\theta _2$, $\psi$ and $\omega$ represent the four adjustable parameters in this scheme.
Now, we will present a scheme for the general single-mode Gaussian operation. Figure 1 shows this schematic diagram with a two-mode entangled state. The nondegenerate optical parametric amplifier (NOPA) is used multiple times in the previous work.[14,15] The output signal and idler modes from the NOPA are two orthogonally polarized modes with the same frequency and the same propagation direction. They can be directly separated by a polarized beam splitter (PBS). Operating under the deamplification condition, the outputs of the NOPA have anticorrelated amplitude quadratures and correlated phase quadratures. By adjusting the angle of the half-wave plate (HWP) after the NOPA cavity, two differently polarized squeezed states can be obtained and separated from the two output ports of the PBS.[20] Based on this case, when the HWP is rotated with an extra angle $\omega /2$, the polarization of one squeezed state is rotated $\omega$ to the vertical direction, and the other squeezed state will be rotated $\omega +\pi$ to the horizontal direction. In other words, the HWP and PBS couple the two squeezed states with an adjustable beam splitter by a splitting ratio dependent on $\omega$. Therefore the output modes from the two output ports of the PBS can be described as $$\begin{alignat}{1} \left(\begin{matrix} {\hat {a}_{\rm E1}} \\ {\hat {a}_{\rm E2}} \\ \end{matrix}\right)=\left(\begin{matrix} {\cos \omega} & {\sin \omega} \\ {\sin \omega} & {-\cos \omega} \\ \end{matrix}\right)\left(\begin{matrix} {\hat {a}_{\rm s1}} \\ {\hat {a}_{\rm s2}} \\ \end{matrix}\right),~~ \tag {2} \end{alignat} $$ where $\hat {a}_{\rm s1} =\hat {x}_{10} e^r+i\hat {p}_{10} e^{-r}$ and $\hat {a}_{\rm s2} =\hat {x}_{20} e^{-r}+i\hat {p}_{20} e^r$ describe the two squeezed states from the NOPA, and $r$ is the squeezing parameter, which is common to refer to $10\log _{10}e^{-2r}$ dB of squeezing. It can be calculated from Eq. (2) that the combined quadrature components of the variable two-mode entangled states $\hat {a}_{\rm E1}$ and $\hat {a}_{\rm E2}$ satisfy the relationships $\hat {\delta}_1 =\hat {x}_{\rm E2} -\hat {x}_{\rm E1} \tan \omega =-\hat {x}_{20} e^{-r}\sec \omega$ and $\hat {\delta}_2 =\hat {p}_{\rm E2} +\hat {p}_{\rm E1} \cot \omega =\hat {p}_{10} e^{-r}\csc \omega$. Obviously, both $\hat {\delta}_1$ and $\hat {\delta}_2$ are adjustable by changing $\omega$. Note that $\omega$ is the first adjustable parameter. In the ideal case of $r\to \infty$, $\hat {\delta}_1 =\hat {\delta}_2 =0$. Next an input state $\hat {a}_{\rm in}$ is coupled to the output mode $\hat {a}_{\rm E1}$ via a 50% beam splitter with phase difference $\psi$, where $\psi$ is the second adjustable parameter. Then two homodyne detections are used to measure the quadrature components of the coupling modes $\hat {a}_{\rm d1}$ and $\hat {a}_{\rm d2}$, respectively. The measurement results $\hat {X}_{\rm d1} (\theta _1)$ and $\hat {X}_{\rm d2} (\theta _2)$ can be expressed as $$\begin{alignat}{1} \!\!\!\!\!\!\hat {X}_{\rm d1} =\,&[\cos\theta _1 (\hat {x}_{\rm in} \cos \psi -\hat {p}_{\rm in} \sin \psi +\hat {x}_{\rm E1}) \\ &+\sin\theta _1 (\hat {p}_{\rm in} \cos \psi +\hat {x}_{\rm in} \sin \psi +\hat {p}_{\rm E1})]/\sqrt 2, \\ \!\!\!\!\!\!\hat {X}_{\rm d2} =\,&[\cos\theta _2 (\hat {x}_{\rm in} \cos \psi -\hat {p}_{\rm in} \sin \psi -\hat {x}_{\rm E1}) \\ &+\sin\theta _2 (\hat {p}_{\rm in} \cos \psi +\hat {x}_{\rm in} \sin \psi -\hat {p}_{\rm E1})]/\sqrt 2,~~ \tag {3} \end{alignat} $$ where $\theta _1$ and $\theta _2$ are the measurement angles of the two balanced homodyne detection systems. Thus including $\theta _1$ and $\theta _2$, there are four adjustable parameters in the measurement results. These parameters can adjust the quadrature components of the input state and the auxiliary entangled state independently. Then the measurement results are fed forward to mode $\hat {a}_{\rm E2}$ with a proper feedforward circuit which is selected to eliminate the noise part from the entangled state. Then, the final output mode is equivalent to a transformed input state. It becomes $$\begin{align} \left(\begin{matrix} {\hat {x}_{\rm out}} \\ {\hat {p}_{\rm out}} \\ \end{matrix}\right)=\,&\left(\begin{matrix} {\hat {x}_{\rm E2}} \\ {\hat {p}_{\rm E2}} \\ \end{matrix}\right)+G\left(\begin{matrix} {\hat {X}_{\rm d1}} \\ {\hat {X}_{\rm d2}} \\ \end{matrix}\right)\\ =\,&L\left(\begin{matrix} {\hat {x}_{\rm in}} \\ {\hat {p}_{\rm in}} \\ \end{matrix}\right)+\left(\begin{matrix} {\hat {\delta}_1} \\ {\hat {\delta}_2} \\ \end{matrix}\right),~~ \tag {4} \end{align} $$ where the feedforward circuit is $$\begin{alignat}{1} G=\frac{\sqrt 2}{t_2 -t_1}\left(\begin{matrix} {t_3 /\sin\theta _1} & {t_3 /\sin\theta _2} \\ {t_2 /(t_3 \sin\theta _1)} & {t_1 /(t_3 \sin\theta _2)} \\ \end{matrix}\right),~~ \tag {5} \end{alignat} $$ and $t_1 =\cot \theta _1$, $t_2 ={\rm cot}\theta _2$, $t_3 ={\rm tan}\omega$ (here $t_1$, $t_2$ and $t_3$ can be taken as any real number). The final transformation matrix $L$ is variable with the four parameters $$\begin{align} \left(\begin{matrix} {\frac{[\cos \psi (t_1 +t_2)+2\sin\psi ]t_3}{t_2 -t_1}} & {\frac{[-2\cos \psi +(t_1 +t_2)\sin\psi ]t_3}{t_1 -t_2}} \\ {\frac{2\cos \psi t_1 t_2 +(t_1 +t_2)\sin\psi}{(t_2 -t_1)t_3}} & {-\frac{\cos \psi (t_1 +t_2)-2t_1 t_2 \sin\psi}{(t_1 -t_2)t_3}} \\ \end{matrix}\right).~~ \tag {6} \end{align} $$ The terms $\sin\psi$ and $\cos \psi$ in the matrix $L$ can be written as a function of $\tan \frac{\psi}{2}$ which can be taken as any real number. The terms $\hat {\delta}_1$ and $\hat {\delta}_2$ in Eq. (4) are the excess noises. When $\theta _1 =0$ ($t_1 \to \infty$ in this case), it is also feasible in practice. The feedforward circuit and transformation matrix $L$ will become their limit values $$\begin{align} G=\,&-\sqrt 2 \left(\begin{matrix} {t_3} & 0 \\ {t_2 /t_3} & {1/(t_3 \sin\theta _2)} \\ \end{matrix}\right),~~ \tag {7} \end{align} $$ $$\begin{align} L=\,&\left(\begin{matrix} {-t_3 \cos \psi} & {t_3 \sin\psi} \\ {-\frac{2t_2 \cos \psi +\sin\psi}{t_3}} & {-\frac{\cos \psi -2t_2 \sin\psi}{t_3}} \\ \end{matrix}\right).~~ \tag {8} \end{align} $$ The general single-mode Gaussian operation only involves three degrees of freedom.[19] Here one can flexibly use a certain parameter as the initial condition. From the corresponding relationships between Eq. (6) and Eq. (1), $t_1$, $t_2$ and $t_3$ can be solved by setting different initial parameters $\psi$. The four parameters are enough for general single-mode Gaussian operation and one solution that covers all cases is classified in Table 1. For example, the shearing operation $D_{2,\hat {x}} (s)=\exp (is\hat {x}^2)$[4,7] corresponds to the transformation of $\alpha =s$, $\beta =\gamma =0$. Therefore the case of $b=0$ in Table 1 is available. In detail, the parameters can be determined as follows: $\psi =0$, $\theta _1 =0$, $t_2 =\frac{s}{2}$ and $t_3 =-1$. The case of $b=0$ under the condition of $d=0$ can be used to realize the squeezing operation $\hat {S}(r')=e^{i{r}'(\hat {x}\hat {p}+\hat {p}\hat {x})}$ with the squeezing parameter ${r}'=\ln a$. It should be noted that Table 1 is not the only solution as the choice of the phase $\psi$ is flexible. For example, the rotation operation $\hat {R}(\theta)=e^{-i\theta (\hat {x}^2+\hat {p}^2)}$ is convenient to choose $\psi =-\theta$, $\theta _1 =0$, $t_2 =0$ and $t_3 =-1$.
Table 1. A corresponding parameter table under different conditions.
Cases $\psi $ $t_1 =\cot \theta _1 $ $t_2 ={\rm cot}\theta _2 $ $t_3 ={\rm tan}\omega $
$b\ne 0$, $c\ne 0$, $ac\geqslant 0$ $\psi =0$ $t_1 =-\frac{a+\sqrt {a/c}}{b}$ $t_2 =\frac{-a+\sqrt {{a/c}}}{b}$ $t_3 =-\sqrt {{a/c}} $
$b\ne 0$, $c\ne 0$, $ac < 0$ $\psi =\pi /2$ $t_1 =\frac{b+b\sqrt {1-ac}}{a\sqrt {1-ac}}$ $t_2 =\frac{b-b\sqrt {1-ac}}{a\sqrt {1-ac}}$ $t_3 =\frac{-b}{\sqrt {1-ac}}$
$b=0$ $\psi =0$ $\theta _1 =0$ $t_2 =\frac{ad}{2}$ $t_3 =-a$
$c=0$, $a\ne 0$ $\psi =\pi /2$ $t_1 =0$ $t_2 =\frac{2b}{a}$ $t_3 =b$
$c=0$, $a=0$ $\psi =0$ $t_1 =-d$ $t_2 =d$ $t_3 =-1$
In practice, we are also concerned about the quality of the realized operations. The fidelity is always used to assess the similarity between the obtained output state $\hat {\rho}_2$ and the ideal output state $\hat {\rho}_1$. It describes the overlap of the two states.[21,22] In the ideal case of infinite squeezing ($r\to \infty$, $\hat {\delta}_1 =\hat {\delta}_2 =0$), it can be seen from Eq. (4) that $\hat {\rho}_2$ turns to $\hat {\rho}_1$ and the fidelity becomes 1. Obviously, they are the excess noise terms that affect the fidelity. The smaller scale entangled state used in our scheme has an advantage to lead small excess noise. Considering that the excess noise terms are related to the parameter $t_3$, it is appropriate to give priority to the parameter $t_3$ to obtain a better output mode. If the quadrature component $\hat {x}_{\rm out}$ (or $\hat {p}_{\rm out}$) of the output mode is more sensitive to the excess noises, one can choose a suitable $t_3$ to make $\sec \omega$ (or $\csc \omega$) smaller. This means that the optimum $t_3$ varies for different cases. Thus when the input state and the transformation matrix are known, it is better to choose the optimum $t_3$ as the initial parameter to determine the remaining three parameters. Let us analyze a specific logic gate. Equation (1) indicates that the transformation matrix of the operation $U=e^{i\frac{\pi}{2}(d\hat {x}^2+\hat {p}^2/d)}$ is $L=\left(\begin{matrix} 0 & {-1/d}\\d & 0 \\\end{matrix}\right)$. This transformation stands for an equivalent cascaded single-mode logic operation which consists of a squeezing operation with squeezing parameter ${r}'=\ln d$ followed by a Fourier transformation. For a vacuum input state, the fidelity of the output mode can be written as $F=2/\sqrt {[e^{-2r}(t_3 ^2+1)+2/d^2][e^{-2r}(1/t_3 ^2+1)+2d^2]}$.[21,22] Figure 2 shows the fidelity of this operation for a vacuum input state. In Figs. 2(a) and 2(b), the solid, dashed and dotted lines describe the fidelity for $d=2$, $d=3$ and $d=4$, respectively. In Fig. 2(a), the fidelity is a function of the parameter $t_3$ with the squeezing parameter $r=0.5$ and different $d$. There exists an optimum $t_3$ to obtain the maximal fidelity obviously. In this case, the optimum $t_3$ can be calculated by $t_3=\frac{(d^2+2e^{2r})^{1/4}}{(d^2+2d^4e^{2r})^{1/4}}$. In Fig. 2(b), the fidelity is a function of the squeezing parameter with the optimum $t_3$ and different $d$. The fidelity for the output state approaches to 1 when the squeezing parameter becomes larger. The operation with parameter $d$ close to 1 is easy to obtain a high fidelity. When the optimum $t_3$ is chosen as the initial parameter, the remaining three parameters can be solved out, which are $\psi =0$, $t_1 =dt_3$, $t_2 =-dt_3$ in detail. Analogously, the optimum $t_3$ for the other operations can be obtained by a similar way.
cpl-34-7-070301-fig2.png
Fig. 2. Fidelity of the given operation for a vacuum input state. (a) The fidelity as a function of the parameter $t_3$ with $r=0.5$ and different $d$. (b) The fidelity as a function of the squeezing parameter with the optimum $t_3$ and different $d$.
In conclusion, we have proposed a scheme to realize the general single-mode Gaussian operation with a two-mode entangled state by adjusting four angle parameters. A certain parameter can be flexibly fixed as the initial condition according to the experimental conditions and the other three adjustable parameters can be solved out. What is more, a two-mode entangled state which is obtained by coupling two independent squeezed states with a beam splitter[23] is also suitable to obtain a similar scheme. In this case, the coupling phase difference between the squeezed states can be selected to replace the parameter $t_3$. In addition, to improve the fidelity of a given case, it is convenient to choose the optimum $t_3$ as the initial condition. This flexible scheme with better results improves the utilization of quantum resources and simplifies the experimental setup. It deserves to be utilized as a basic efficient element in the future large-scale quantum computation. We thank Professor Xiaolong Su for the useful discussions. References One-Way Quantum Computation with Cluster State and Probabilistic GateOptical logic gates using coherent feedbackQuantum computing with continuous-variable clustersQuantum simulation of quantum field theory using continuous variablesA One-Way Quantum ComputerUniversal Quantum Computation with Continuous-Variable Cluster StatesContinuous-variable Gaussian analog of cluster statesBuilding Gaussian cluster states by linear opticsGaussian quantum informationToward demonstrating controlled-X operation based on continuous-variable four-partite cluster states and quantum teleportersDemonstration of Unconditional One-Way Quantum Computations for Continuous VariablesDemonstration of a Controlled-Phase Gate for Continuous-Variable One-Way Quantum ComputationContinuous variable quantum optical simulation for time evolution of quantum harmonic oscillatorsGates for one-way quantum computation based on Einstein-Podolsky-Rosen entanglementGate sequence for continuous variable one-way quantum computationExperimental preparation of eight-partite cluster state for photonic qumodesScheme for preparation of multi-partite entanglement of atomic ensemblesUniversal linear Bogoliubov transformations through one-way quantum computationExperimental generation of bright two-mode quadrature squeezed light from a narrow-band nondegenerate optical parametric amplifierDistinguishing two single-mode Gaussian states by homodyne detection: An information-theoretic approachFidelity for displaced squeezed thermal states and the oscillator semigroupGeneration of continuous-wave broadband entangled beams using periodically poled lithium niobate waveguides
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