用IWOP技术和纠缠态表象发展量子相空间理论
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摘要
量子相空间理论从玻尔-索末菲量子化发展到Wigner提出的对应于密度算符的准几率分布函数W (q, p)，成为一个里程碑。W(q, p)避免了由于海森堡不确定原理（不能同时精确地测量微观粒子的位置q和动量p）所带来的不能定义(q-p)函数的问题，因此Wigner函数不是真正意义上的几率分布，但是Wigner函数的边缘分布分别给出在坐标空间和动量空间测量到粒子的几率，因此有广泛的应用。本文用有序算符内的积分技术（IWOP技术）以崭新的视角和方法研究量子相空间中准几率分布函数表示、重构、以及经典函数量子化、经典变换与量子幺正变换的关系等。主要内容如下：
一、利用IWOP技术揭示Weyl编序的实质，并在此基础上引入Weyl编序算符内的积分技术，明显地发展Weyl对应与Wigner算符理论，给出Wigner算符的Weyl编序函数形式以用于发展密度算符的量子相空间描述。在纠缠情况下，谈单个粒子的量子态没有物理意义，我们引入纠缠形式的Wigner算符来描写它，因为此时纠缠Wigner函数的边缘分布反映的是整个系统所处的态包含的纠缠性质。最后将Weyl变换推广到两模情形。
二、我们发现相干态在量子相空间的代表小面积的运动受量子刘维定理的支配，发现它对应于经典光学中的菲涅尔变换，从而可以定义菲涅尔算符，然后我们讨论菲涅尔算符与Wigner算符转动之间的关系。发现菲涅尔变换的积分核恰好是坐标-动量中介表象与坐标表象的内积，即找到了经典光学中菲涅尔变换的量子对应。作为应用，讨论菲涅尔变换与量子断层摄影理论的关系，以及其在求解哈密顿量本征函数中的应用。提出并构造双模菲涅尔算符，把以上讨论推广到纠缠情况。量子力学和光学的比拟最早有薛定谔指出，也是他发现薛定谔方程的思路。我们的理论符合薛定谔的思路。
三、在相空间引入一种新的积分变换（为了方便简称为新变换）并研究其相关特性。它良好的变换特性可用于研究Weyl编序和P?Q (Q?P)编序之间的关系，从光学角度看，利用此变换可以由―啁啾‖函数导出分数傅里叶变换积分核。对于纠缠情况我们也给出了这类新变换具体的讨论，并研究其相关性质与应用。我们期望能在光学上实现这一类新变换。
四、通过引入s参数化的Wigner算符，我们将IWOP技术推广到s-编序算符内的积分技术（IWSOP技术），它是一种综合了正规编序，反正规编序和Weyl编序的积分技术，并导出了密度算符的s-编序展开公式，讨论算符s-编序的本质，引入带s参数的新变换，并将量子光学中的光子计数公式推广到带s参数的形式，这些内容都极大地便利了对量子相空间理论的研究。最后是对双模情况下s参数Wigner算符及其s-编序展开公式的介绍。
五、利用IWOP技术我们将连续纠缠态表象推广到三粒子情形，研究其压缩、制备，并在此基础上引入三模Wigner算符以及对其在量子隐态传输、量子断层摄影理论中具体应用的讨论。最后，基于两个相互共轭的三粒子纠缠态表象，我们用两种方法得到了三模复分数傅立叶变换，并证明其具有群的乘法性质，通过定义三变量厄密多项式找到其本征模，最后是对其卷积定理的讨论。
The Quantum phase space theory becomes a Milestone from the Bohr-Sommerfeld quantization developing to the quasi-probability distribution function W q ,p of density operator proposed by Wigner. W q ,p avoids the problem of undefining of the q ,p function casued by Heisenberg uncertainty principle (we can not measure the position q and momentun p of the particle exactly), thus the Wigner function is not a real probability distribution. However, the marginal distributions of the Wigner function give the probability of measuring the positionq and momentun p of the particle respectively. By virtue of the technique of integration within anQ ordered product (IWOP) of operators, the paper study the representation, reconstruction of the quasi-probability distribution function and quantization of classical function, the relationship between classical transform and quantum unitary transform ect.. Our method is novel and unique. The main content of the paper is:
1. By virtue of the IWOP techique, we reveal the essence of the Weyl ordering. By introducing the technique of integration within a Weyl ordered product (IWWOP) of operators, we obtain the Weyl ordered form of the Wigner operator which can be applied to develop the Quantum phase space theory. It can be seen from the article that the rule of the Weyl correspondence and the Wigner operator theory will be developed explicitly. On the other hand, we emphasize that for entangled particles one should treat their wave function as a whole, there is no physical meaning for an isolated particle's Wigner function, therefore thinking of entangled Wigner function (Wigner operator) is of necessity. Finally, we generalize the Weyl transform into entangled case.
2. The coherent state can be represented by a small area in quantum phase space. We find that the move of the small area is controlled by the Liouville Theorem and is correspondence to the Fresnel transform in classical optics. Thus we can define the Fresne operator and discuss its association with the Wigner transform, i.e., there exists algorithmic isomorphism between ABCD transform of the Wigner distribution function and the optical Fresnel transform. We also find the kernel of the Fresnel transform is the inner product of the intermediate coordinate-momentum representation and coordinate representation, that is to say, we find the quantum correspondence of the Fresnel transform in classical optics. Additionally, we apply the Fresnel operator to discuss its relationship with the quantum tomograghy, and also to solve the Hamiltonian equation.Finally, we propose and construct the two-mode Fresnel operator and extend the above discussions to entangled case. The analogy of quantum mechanics and optics was pointed out by Schr?dinger. It is the analogy that he finds the idea of Schr?dinger equation. Our theory is consistent with Schr?dinger's ideas.
3. We propose a new two-fold integration transform in q pphase space (we call it new transform in conveniece) which possesses some well-behaved transform properties and can be applied to finding the connection between the Weyl ordering and P Q (Q P ) ordering of operators. According to this new transform, we can obtain the fractional Fourier transform kernel from the chirplet function. We also develop this kind new transform to a more general case that can be further related to the transform between two mutually conjugate entangled state representations and study its properties and applications. We expect this transform could be implemented by experimentalists.
4. By introducing the s-parameterized Wigner operator, we generalize the IWOP technique to the technique of integration within the s-ordered product of operators (IWSOP) which can unify the three integration techniques (within normal-, Weyl- and antinormal-ordering of operators) as one, and derive the s-ordered operator expansion formula of density operators. On the other hand, the essence of the s-parameterized quantization is discussed, the s-parameterized new transform is obtained and the photocount formula is also generalized into the s-parameterized case which can reduc to the usual one when s takes particular values. Our discussions provide greatly convenience for the study of the Quantum phase space theory. Finally, we extend the above discussions to the entangled case.
5. By virtue of the IWOP technique, we correctly construct the tripartite entangled state representations (TESR) and investigate its quantum properties and some applications. We derive the three-mode squeezed operator and introduce the optical network for generating such an ideal tripartite entangled state. The three-mode entangled Wigner operator in this representation is constructed. Based on this form, we calculate the Wigner function expressions of some tripartite entangled states. The path integral formalism related to the tripartite entangled state is demonstrated. Additionally, the application of such entangled state in quantum teleportation and quantum Tomography thoery is analyzed as well. In the end, we introduce the three-mode optical entangled fractional Fourier transform (EFrFT) through two methods. The EFrFT, which is characteristic of the eigenmodes being three-variable Hermite polynomials, satisfies the additivity property. We also define two functions' convolution in the EFrFT scheme and obtain the convolution theorem using the TESR. The derivation is concise and rigorous because our calculation is based on Dirac's powerful representation theory.
一、利用IWOP技术揭示Weyl编序的实质，并在此基础上引入Weyl编序算符内的积分技术，明显地发展Weyl对应与Wigner算符理论，给出Wigner算符的Weyl编序函数形式以用于发展密度算符的量子相空间描述。在纠缠情况下，谈单个粒子的量子态没有物理意义，我们引入纠缠形式的Wigner算符来描写它，因为此时纠缠Wigner函数的边缘分布反映的是整个系统所处的态包含的纠缠性质。最后将Weyl变换推广到两模情形。
二、我们发现相干态在量子相空间的代表小面积的运动受量子刘维定理的支配，发现它对应于经典光学中的菲涅尔变换，从而可以定义菲涅尔算符，然后我们讨论菲涅尔算符与Wigner算符转动之间的关系。发现菲涅尔变换的积分核恰好是坐标-动量中介表象与坐标表象的内积，即找到了经典光学中菲涅尔变换的量子对应。作为应用，讨论菲涅尔变换与量子断层摄影理论的关系，以及其在求解哈密顿量本征函数中的应用。提出并构造双模菲涅尔算符，把以上讨论推广到纠缠情况。量子力学和光学的比拟最早有薛定谔指出，也是他发现薛定谔方程的思路。我们的理论符合薛定谔的思路。
三、在相空间引入一种新的积分变换（为了方便简称为新变换）并研究其相关特性。它良好的变换特性可用于研究Weyl编序和P?Q (Q?P)编序之间的关系，从光学角度看，利用此变换可以由―啁啾‖函数导出分数傅里叶变换积分核。对于纠缠情况我们也给出了这类新变换具体的讨论，并研究其相关性质与应用。我们期望能在光学上实现这一类新变换。
四、通过引入s参数化的Wigner算符，我们将IWOP技术推广到s-编序算符内的积分技术（IWSOP技术），它是一种综合了正规编序，反正规编序和Weyl编序的积分技术，并导出了密度算符的s-编序展开公式，讨论算符s-编序的本质，引入带s参数的新变换，并将量子光学中的光子计数公式推广到带s参数的形式，这些内容都极大地便利了对量子相空间理论的研究。最后是对双模情况下s参数Wigner算符及其s-编序展开公式的介绍。
五、利用IWOP技术我们将连续纠缠态表象推广到三粒子情形，研究其压缩、制备，并在此基础上引入三模Wigner算符以及对其在量子隐态传输、量子断层摄影理论中具体应用的讨论。最后，基于两个相互共轭的三粒子纠缠态表象，我们用两种方法得到了三模复分数傅立叶变换，并证明其具有群的乘法性质，通过定义三变量厄密多项式找到其本征模，最后是对其卷积定理的讨论。
The Quantum phase space theory becomes a Milestone from the Bohr-Sommerfeld quantization developing to the quasi-probability distribution function W q ,p of density operator proposed by Wigner. W q ,p avoids the problem of undefining of the q ,p function casued by Heisenberg uncertainty principle (we can not measure the position q and momentun p of the particle exactly), thus the Wigner function is not a real probability distribution. However, the marginal distributions of the Wigner function give the probability of measuring the positionq and momentun p of the particle respectively. By virtue of the technique of integration within anQ ordered product (IWOP) of operators, the paper study the representation, reconstruction of the quasi-probability distribution function and quantization of classical function, the relationship between classical transform and quantum unitary transform ect.. Our method is novel and unique. The main content of the paper is:
1. By virtue of the IWOP techique, we reveal the essence of the Weyl ordering. By introducing the technique of integration within a Weyl ordered product (IWWOP) of operators, we obtain the Weyl ordered form of the Wigner operator which can be applied to develop the Quantum phase space theory. It can be seen from the article that the rule of the Weyl correspondence and the Wigner operator theory will be developed explicitly. On the other hand, we emphasize that for entangled particles one should treat their wave function as a whole, there is no physical meaning for an isolated particle's Wigner function, therefore thinking of entangled Wigner function (Wigner operator) is of necessity. Finally, we generalize the Weyl transform into entangled case.
2. The coherent state can be represented by a small area in quantum phase space. We find that the move of the small area is controlled by the Liouville Theorem and is correspondence to the Fresnel transform in classical optics. Thus we can define the Fresne operator and discuss its association with the Wigner transform, i.e., there exists algorithmic isomorphism between ABCD transform of the Wigner distribution function and the optical Fresnel transform. We also find the kernel of the Fresnel transform is the inner product of the intermediate coordinate-momentum representation and coordinate representation, that is to say, we find the quantum correspondence of the Fresnel transform in classical optics. Additionally, we apply the Fresnel operator to discuss its relationship with the quantum tomograghy, and also to solve the Hamiltonian equation.Finally, we propose and construct the two-mode Fresnel operator and extend the above discussions to entangled case. The analogy of quantum mechanics and optics was pointed out by Schr?dinger. It is the analogy that he finds the idea of Schr?dinger equation. Our theory is consistent with Schr?dinger's ideas.
3. We propose a new two-fold integration transform in q pphase space (we call it new transform in conveniece) which possesses some well-behaved transform properties and can be applied to finding the connection between the Weyl ordering and P Q (Q P ) ordering of operators. According to this new transform, we can obtain the fractional Fourier transform kernel from the chirplet function. We also develop this kind new transform to a more general case that can be further related to the transform between two mutually conjugate entangled state representations and study its properties and applications. We expect this transform could be implemented by experimentalists.
4. By introducing the s-parameterized Wigner operator, we generalize the IWOP technique to the technique of integration within the s-ordered product of operators (IWSOP) which can unify the three integration techniques (within normal-, Weyl- and antinormal-ordering of operators) as one, and derive the s-ordered operator expansion formula of density operators. On the other hand, the essence of the s-parameterized quantization is discussed, the s-parameterized new transform is obtained and the photocount formula is also generalized into the s-parameterized case which can reduc to the usual one when s takes particular values. Our discussions provide greatly convenience for the study of the Quantum phase space theory. Finally, we extend the above discussions to the entangled case.
5. By virtue of the IWOP technique, we correctly construct the tripartite entangled state representations (TESR) and investigate its quantum properties and some applications. We derive the three-mode squeezed operator and introduce the optical network for generating such an ideal tripartite entangled state. The three-mode entangled Wigner operator in this representation is constructed. Based on this form, we calculate the Wigner function expressions of some tripartite entangled states. The path integral formalism related to the tripartite entangled state is demonstrated. Additionally, the application of such entangled state in quantum teleportation and quantum Tomography thoery is analyzed as well. In the end, we introduce the three-mode optical entangled fractional Fourier transform (EFrFT) through two methods. The EFrFT, which is characteristic of the eigenmodes being three-variable Hermite polynomials, satisfies the additivity property. We also define two functions' convolution in the EFrFT scheme and obtain the convolution theorem using the TESR. The derivation is concise and rigorous because our calculation is based on Dirac's powerful representation theory.
引文
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[1] Fan Hongyi. Symplectic Group Representation of the Two-Mode Squeezing Operator in the Coherent State Basis [J]. Commun. Theor. Phys., 2003, 40: 589-594.
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[1] Fan Hongyi. A New Kind of Two-Fold Integration Transformation in Phase Space and Its Uses in Weyl Ordering of Operators [J]. Commun. Theor. Phys., 2008, 50: 935-937.
[2] Fan Hongyi. An approach for converting Weyl ordered operators to Q-P ordering and P-Q ordering [J]. to be published.
[3] Fan Hongyi, Hu Liyun. New relation between input and output fields based on Collins diffraction formula and the Wigner function in entangled state representation [J]. J. Mod. Opt., 2009, 56: 1819-1823.
[4]Lv Cuihong, Fan Hongyi. A New Kind of Integration Transformation in Phase Space Related to Two Mutually Conjugate Entangled-State Representations and Its Uses in Weyl Ordering of Operators [J]. Chin. Phys. Lett., 2010, 27: 050301.
[5]Fan Hongyi, Lv Cuihong. New complex integration transformation and its compatibility with complex Weyl-Wigner transformation and entangled state representation [J]. J. Opt. Soc. Am. A, 2009, 26: 1.
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[1] Hu Liyun, Lu Hailiang. Application of three-mode Einstein-Podolsky-Rosen entangled state with continuous variables to teleportation [J]. Chin. Phys., 2007,16: 2200.
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[12] Lv Cuihong, Fan Hongyi. to be published.
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[18] Chountasis S, Vourdas A, Bendjaballah C. Fractional Fourier operators and generalized Wigner functions [J]. Phys. Rev. A, 1999, 60: 3467-3473.
[19] Fan Hongyi, Fan Yue. EPR entangled states and complex fractional Fourier transformation [J]. Eur. Phys. J. D, 2002, 21: 233-238.
[20]Lu CuiHong, Fan Hongyi, Jiang Nianquan. Two mutually conjugated tripartite entangled states and their fractional Fourier transformation kernel [J].Chin. Phys. B, 2010, 19:120303.
[21]Lv Cuihong, Fan Hongyi. Optical entangled fractional Fourier transform derived via non-unitary SU(2) bosonic operator realization and its convolution theorem, [J]. Opt. Commun., 2011, 28: 1925-1932.
[22]Brian G. Wybourne. Classical Groups for Physicists [M]. New York, 1974.
[23]Liu Shuguang, Fan Hongyi. Convolution theorem of 3-dimensional entangled fractional Fourier transformation deduced by the tripartite entangled state representation [J]. Theor. Math. Phys., 2009, 161: 1710-1718.
[24]Fan Hongyi, Jiang Nianquan. On the Entangled Fractional Fourier Transform in Tripartite Entangled State Representation [J]. Commun. Theor. Phys., 2003, 40: 39 44; Qian Xiaoqing, Song Tongqiang. Entangled Fractional Fourier Transform for the Multipartite Entangled State Representation [J]. Commun. Theor. Phys., 2006, 45: 1097-1101.
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[1] Fan Hongyi. Applications of Weyl Ordered Two-Mode Wigner Operator for Quantum Mechanical Entangled System [J]. Commun. Theor. Phys., 2004, 41: 205-208; Weyl-ordered Polynomials Studied by virtue of the IWWOP technique [J]. Mod. Phys. Lett. A, 2000, 15: 2297-2303; Weyl ordering quantum mechanical operators by virtue of the IWOP technique [J]. J. Phys. A, 1992, 25: 3443-3447; Fan Hongyi, Fan Yue. Weyl Ordering for Entangled State Representation [J]. Intern. J. Mod. Phys. A, 2002, 17: 701-708.
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[4] Lv Cuihong, Fan Hongyi. Weyl orering technique for converting energy-eigenstates equation to the equation for Wigner function of eigenstates [J]. to be published.
[5]范洪义.从量子力学到量子光学-数理进展[M].上海：上海交通大学出版社, 2005.
[1] Fan Hongyi. Symplectic Group Representation of the Two-Mode Squeezing Operator in the Coherent State Basis [J]. Commun. Theor. Phys., 2003, 40: 589-594.
[2] Fan Hongyi, Lu Hailiang. Wave-function transformations by general SU(1,1) single-mode squeezing and analogy to Fresneltransformations in wave optics [J]. Opt. Commun., 2006, 258: 51-58.
[3] Lv Cui-hong, Fan Hong-yi. Adaption of optical Fresnel transform to optical Wigner transform [J]. Phys. Scr., 2010, 82: 025004.
[4]Lohmann A W. Image rotation, Wigner rotation, and the fractional Fourier transform [J]. J. Opt. Soc. Am. A, 1993, 10: 2181-2186.
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[8] Fan Hongyi,Guo Qin. Quantum phase space theory based on Intermediate Coordinate-Momentum representation [J]. Mod. Phys. Lett. B, 2007, 27: 1831-1836.
[9] Lv Cui-hong, Fan Hong-yi. Application of the intermediate coordinate-momentum representation [J]. to be published.
[10] Fan Hongyi, Lu Hailiang. 2-mode Fresnel operator and entangled Fresnel transform [J]. Phys. Lett. A, 2005, 334: 132-139.
[11] Fan Hongyi, Chen Junhua. Coherent State Projection Operator Representation of Symplectic Transformations as a Loyal Representation of Symplectic Group [J]. Commun. Theor. Phys., 2002, 38: 147-150.
[1] Fan Hongyi. A New Kind of Two-Fold Integration Transformation in Phase Space and Its Uses in Weyl Ordering of Operators [J]. Commun. Theor. Phys., 2008, 50: 935-937.
[2] Fan Hongyi. An approach for converting Weyl ordered operators to Q-P ordering and P-Q ordering [J]. to be published.
[3] Fan Hongyi, Hu Liyun. New relation between input and output fields based on Collins diffraction formula and the Wigner function in entangled state representation [J]. J. Mod. Opt., 2009, 56: 1819-1823.
[4]Lv Cuihong, Fan Hongyi. A New Kind of Integration Transformation in Phase Space Related to Two Mutually Conjugate Entangled-State Representations and Its Uses in Weyl Ordering of Operators [J]. Chin. Phys. Lett., 2010, 27: 050301.
[5]Fan Hongyi, Lv Cuihong. New complex integration transformation and its compatibility with complex Weyl-Wigner transformation and entangled state representation [J]. J. Opt. Soc. Am. A, 2009, 26: 1.
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[2] Fan Hongyi. Unifying the theory of Integration within normal-, Weyl- and antinormal-ordering of operators and the s-ordered operator expansion formula of density operators [J]. Chin. Phys. B, 2010, 19: 050303.
[3] Fan Hongyi, Lv Cuihong. s-Parameterized phase space distribution for describing no counts registered on a photonic detector, Opt. Commun. 2009, 282: 4741-4744.
[4] Fan Hongyi, Yuan Hongchun, Hu Liyun. The essence of the s-parameterized quantization scheme [J]. to be published.
[5] Lv Cuihong, Fan Hongyi. A new two-fold s-parameterized integration transformation [J]. to be published.
[6]Lv Cuihong, Yuan Hongchun, Fan Hongyi. A new general s-parameterized photocount formula [J]. to be published.
[7] Fan Hongyi, Hu Liyun, Yuan Hongchun. S-parameterized Weyl-Wigner correspondence in the entangled form and its applications [J]. Chin. Phys. B, 2010, 19: 060305.
[1] Hu Liyun, Lu Hailiang. Application of three-mode Einstein-Podolsky-Rosen entangled state with continuous variables to teleportation [J]. Chin. Phys., 2007,16: 2200.
[2] Fan Hongyi, Sun Mingzai. Bipartite entangled states and its application in quantum communication [J]. J. Opt. B: Quantum Semiclass. Opt., 2002, 4: 228-234.
[3] Fan Hongyi, Fan Yue. New approach for solving master equations of density operators by virtue of the thermal entangled states [J]. J. Phys. A: Math. Gen. 2002, 35: 6873-6882.
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[5] Fan Hongyi, Jiang Nianquan. Tripartite entangled wigner operator, the Wigner function and its marginal distributions [J]. J. Opt. B: Quantum Semiclass. Opt., 2003, 5: 283-288.
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[7] Lv Cuihong, Yuan Hongchun, Hong-yi Fan. Mutually conjugate tripartite entangled state representations: construction and applications [J].to be published.
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[11] Lv Cuihong, Fan Hongyi. A Kind Tripartite Entangled State Representation and Its Application in Quantum Teleportation [J]. Int J Theor Phys 2010, 49: 1944 1951.
[12] Lv Cuihong, Fan Hongyi. to be published.
[13] Vogel K, Risken H. Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase [J]. Phys. Rev. A, 1989, 40: 2847-2849.
[14] Smithey D T, Beck M, Raymer M G, Faridani A. Multifragment azimuthal correlation functions: Probes for reaction dynamics in collisions of intermediate energy heavy ions [J]. Phys. Rev. Lett. 1993, 70: 1224-1227.
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[16] Mendlovic D, Ozaktas H M. Fractional Fourier transforms and their optical implementation:1 [J]. J. Opt. Soc. Am. A, 1993, 10: 1875-1881;
[17] Lohmann A W. Image rotation, Wigner rotation and the fractional Fourier transform[J].J. Opt. Soc. Am. A, 1993, 10: 2181-2186.
[18] Chountasis S, Vourdas A, Bendjaballah C. Fractional Fourier operators and generalized Wigner functions [J]. Phys. Rev. A, 1999, 60: 3467-3473.
[19] Fan Hongyi, Fan Yue. EPR entangled states and complex fractional Fourier transformation [J]. Eur. Phys. J. D, 2002, 21: 233-238.
[20]Lu CuiHong, Fan Hongyi, Jiang Nianquan. Two mutually conjugated tripartite entangled states and their fractional Fourier transformation kernel [J].Chin. Phys. B, 2010, 19:120303.
[21]Lv Cuihong, Fan Hongyi. Optical entangled fractional Fourier transform derived via non-unitary SU(2) bosonic operator realization and its convolution theorem, [J]. Opt. Commun., 2011, 28: 1925-1932.
[22]Brian G. Wybourne. Classical Groups for Physicists [M]. New York, 1974.
[23]Liu Shuguang, Fan Hongyi. Convolution theorem of 3-dimensional entangled fractional Fourier transformation deduced by the tripartite entangled state representation [J]. Theor. Math. Phys., 2009, 161: 1710-1718.
[24]Fan Hongyi, Jiang Nianquan. On the Entangled Fractional Fourier Transform in Tripartite Entangled State Representation [J]. Commun. Theor. Phys., 2003, 40: 39 44; Qian Xiaoqing, Song Tongqiang. Entangled Fractional Fourier Transform for the Multipartite Entangled State Representation [J]. Commun. Theor. Phys., 2006, 45: 1097-1101.
[25]Ahmed I. Zayed. A Convolution and Product Theorem for the Fractional Fourier Transform [J]. IEEE Singal Process. Lett., 1998, 5: 101-103.