We mostly investigate two schemes. One is to teleport a multi-mode W-type entangled coherent state using a peculiar bipartite entangled state as the quantum channel different from other proposals. Based on our formali...We mostly investigate two schemes. One is to teleport a multi-mode W-type entangled coherent state using a peculiar bipartite entangled state as the quantum channel different from other proposals. Based on our formalism,teleporting multi-mode coherent state or squeezed state is also possible. Another is that the tripartite entangled state is used as the quantum channel of controlled teleportation of an arbitrary and unknown continuous variable in the case of three participators.展开更多
Based on our previous paper (Commun.Theor.Phys.39 (2003) 417) we derive the convolution theoremof fractional Fourier transformation in the context of quantum mechanics,which seems a convenient and neat way.Generalizat...Based on our previous paper (Commun.Theor.Phys.39 (2003) 417) we derive the convolution theoremof fractional Fourier transformation in the context of quantum mechanics,which seems a convenient and neat way.Generalization of this method to the complex fractional Fourier transformation case is also possible.展开更多
For a field k and two finite groups G and X, when G acts on X from the right by group automorphisms, there is a Hopf algebra structure on k-space （kX°P）＊ kG （see Theorem 2.1）, called a non-balanced quantum ...For a field k and two finite groups G and X, when G acts on X from the right by group automorphisms, there is a Hopf algebra structure on k-space （kX°P）＊ kG （see Theorem 2.1）, called a non-balanced quantum double and denoted by Dx（G）. In this paper, some Hopf algebra properties of Dx （G） are given, the representation types of Dx （G） viewed as a k-algebra are discussed, the algebra structure and module category over Dx（G） are studied. Since the Hopf algebra structure of non-balanced quantum double DX （G） generMizes the usual quantum double D（G） for a finite group G, all results about Dx（G） in this paper can also be used to describe D（G） as a special case and the universal R-matrix of Dx （G） provides more solutions of Yang-Baxter equation.展开更多
Majorana's stellar representation provides an intuitive picture in which quantum states in highdimensional Hilbert space can be observed using the trajectory of Majorana stars.We consider the Majorana's stella...Majorana's stellar representation provides an intuitive picture in which quantum states in highdimensional Hilbert space can be observed using the trajectory of Majorana stars.We consider the Majorana's stellar representation of the quantum geometric tensor for a spin state up to spin-3/2.The real and imaginary parts of the quantum geometric tensor,corresponding to the quantum metric tensor and Berry curvature,are therefore obtained in terms of the Majorana stars.Moreover,we work out the expressions of quantum geometric tensor for arbitrary spin in some important cases.Our results will benefit the comprehension of the quantum geometric tensor and provide interesting relations between the quantum geometric tensor and Majorana's stars.展开更多
We construct a fermion analogue of the Fock representation of quantum toroidal algebra and construct the fermion representation of quantum toroidal algebra on the K-theory of Hilbert scheme.
To consummate the quantum pendulum theory whose Hamiltonian takes bosonic operator formalism and manifestly exhibits its dynamic behaviour in the entangled state representation, we introduce angular momentum state rep...To consummate the quantum pendulum theory whose Hamiltonian takes bosonic operator formalism and manifestly exhibits its dynamic behaviour in the entangled state representation, we introduce angular momentum state representation and phase state representation. It turns out that the angular momentum state is the partial wave expansion of the entangled state.展开更多
In this paper,we present three types of representations over Anq defined based on the quantum torus of rank n,which are closely related to modules over some vertex algebras.The isomorphism classes among these modules ...In this paper,we present three types of representations over Anq defined based on the quantum torus of rank n,which are closely related to modules over some vertex algebras.The isomorphism classes among these modules are also determined.展开更多
This paper constructs the new common eigenvectors of n intermediate coordinate-momentum operators which are complete and orthonormal. The intermediate coordinate-momentum representation of a multi-particles system is ...This paper constructs the new common eigenvectors of n intermediate coordinate-momentum operators which are complete and orthonormal. The intermediate coordinate-momentum representation of a multi-particles system is proposed and applied to a generaln-mode quantum harmonic oscillators system with coordinate-momentum coupling.展开更多
With the rapid development of machine learning,artificial neural networks provide a powerful tool to represent or approximate many-body quantum states.It was proved that every graph state can be generated by a neural ...With the rapid development of machine learning,artificial neural networks provide a powerful tool to represent or approximate many-body quantum states.It was proved that every graph state can be generated by a neural network.Here,we introduce digraph states and explore their neural network representations(NNRs).Based on some discussions about digraph states and neural network quantum states(NNQSs),we construct explicitly an NNR for any digraph state,implying every digraph state is an NNQS.The obtained results will provide a theoretical foundation for solving the quantum manybody problem with machine learning method whenever the wave-function is known as an unknown digraph state or it can be approximated by digraph states.展开更多
The quantum many-body problem(QMBP) has become a hot topic in high-energy physics and condensed-matter physics. With an exponential increase in the dimensions of Hilbert space, it becomes very challenging to solve the...The quantum many-body problem(QMBP) has become a hot topic in high-energy physics and condensed-matter physics. With an exponential increase in the dimensions of Hilbert space, it becomes very challenging to solve the QMBP, even with the most powerful computers. With the rapid development of machine learning, artificial neural networks provide a powerful tool that can represent or approximate quantum many-body states. In this paper, we aim to explicitly construct the neural network representations of hypergraph states. We construct the neural network representations for any k-uniform hypergraph state and any hypergraph state,respectively, without stochastic optimization of the network parameters. Our method constructively shows that all hypergraph states can be represented precisely by the appropriate neural networks introduced in [Science 355(2017) 602] and formulated in [Sci. China-Phys.Mech. Astron. 63(2020) 210312].展开更多
A beam splitter operator is a very important linear device in the field of quantum optics and quantum information.It can not only be used to prepare complete representations of quantum mechanics,entangled state repres...A beam splitter operator is a very important linear device in the field of quantum optics and quantum information.It can not only be used to prepare complete representations of quantum mechanics,entangled state representation,but it can also be used to simulate the dissipative environment of quantum systems.In this paper,by combining the transform relation of the beam splitter operator and the technique of integration within the product of the operator,we present the coherent state representation of the operator and the corresponding normal ordering form.Based on this,we consider the applications of the coherent state representation of the beam splitter operator,such as deriving some operator identities and entangled state representation preparation with continuous-discrete variables.Furthermore,we extend our investigation to two single and two-mode cascaded beam splitter operators,giving the corresponding coherent state representation and its normal ordering form.In addition,the application of a beam splitter to prepare entangled states in quantum teleportation is further investigated,and the fidelity is discussed.The above results provide good theoretical value in the fields of quantum optics and quantum information.展开更多
We provide a direct proof of the Seidel representation in the quantum K-theory QK(Gr(k,n))by studying projected Gromov-Witten varieties concretely.As applications,we give an alternative proof of the K-theoretic quantu...We provide a direct proof of the Seidel representation in the quantum K-theory QK(Gr(k,n))by studying projected Gromov-Witten varieties concretely.As applications,we give an alternative proof of the K-theoretic quantum Pieri rule by Buch and Mihalcea(2011),reduce certain quantum Schubert structure constants of higher degree to classical Littlewood-Richardson coefficients for K(Gr(k,n)),and provide a quantum Littlewood-Richardson rule for QK(Gr(3,n)).展开更多
As a part of quantum image processing, quantum image scaling is a significant technology for the development of quantum computation. At present, most of the quantum image scaling schemes are based on grayscale images,...As a part of quantum image processing, quantum image scaling is a significant technology for the development of quantum computation. At present, most of the quantum image scaling schemes are based on grayscale images, with relatively little processing for color images. This paper proposes a quantum color image scaling scheme based on bilinear interpolation, which realizes the 2^(n_(1)) × 2^(n_(2)) quantum color image scaling. Firstly, the improved novel quantum representation of color digital images(INCQI) is employed to represent a 2^(n_(1)) × 2^(n_(2)) quantum color image, and the bilinear interpolation method for calculating pixel values of the interpolated image is presented. Then the quantum color image scaling-up and scaling-down circuits are designed by utilizing a series of quantum modules, and the complexity of the circuits is analyzed.Finally, the experimental simulation results of MATLAB based on the classical computer are given. The ultimate results demonstrate that the complexities of the scaling-up and scaling-down schemes are quadratic and linear, respectively, which are much lower than the cubic function and exponential function of other bilinear interpolation schemes.展开更多
We use the Ringel-Hall algebra approach to study the canonical basis elements for the quantum group of type B2 which are characterized in Xi [12]. However, our approach simplifies several computations there.
The entanglement dynamics of two-qubit systems in different quantum noises are investigated by means of the operator-sum representation method. We find that, except for the amplitude damping and phase damping quantum ...The entanglement dynamics of two-qubit systems in different quantum noises are investigated by means of the operator-sum representation method. We find that, except for the amplitude damping and phase damping quantum noise, the sudden death of entanglement is always observed in different two-qubit systems with generalized amplitude damping and depolarizing quantum noise.展开更多
Let H be a finite Hopf C^＊ -algebra and H′be its dual Hopf algebra. Drinfeld＇s quantum double D（H） of H is a Hopf^＊-algebra. There is a faithful positive linear functional θ on D（H）. Through the associated Ge...Let H be a finite Hopf C^＊ -algebra and H′be its dual Hopf algebra. Drinfeld＇s quantum double D（H） of H is a Hopf^＊-algebra. There is a faithful positive linear functional θ on D（H）. Through the associated Gelfand-Naimark-Segal （GNS） representation, D（H） has a faithful^＊ -representation so that it becomes a Hopf C^＊ -algebra. The canonical embedding map of H into D（H） is isometric.展开更多
We present a semiclassical （SC） approach for quantum dissipative dynamics, constructed on basis of the hierarchical-equation-of-motion （HEOM） formalism. The dynamical components considered in the developed SC-HEOM...We present a semiclassical （SC） approach for quantum dissipative dynamics, constructed on basis of the hierarchical-equation-of-motion （HEOM） formalism. The dynamical components considered in the developed SC-HEOM are wavepackets＇ phase-space moments of not only the primary reduced system density operator but also the auxiliary density operators （ADOs） of HEOM. It is a highly numerically efficient method, meanwhile taking into account the high-order effcts of system-bath couplings. The SC-HEOM methodology is exemplified in this work on the hierarchical quantum master equation [J. Chem. Phys. 131, 214111 （2009）] and numerically demonstrated on linear spectra of anharmonic oscillators.展开更多
We redesign the parameterized quantum circuit in the quantum deep neural network, construct a three-layer structure as the hidden layer, and then use classical optimization algorithms to train the parameterized quantu...We redesign the parameterized quantum circuit in the quantum deep neural network, construct a three-layer structure as the hidden layer, and then use classical optimization algorithms to train the parameterized quantum circuit, thereby propose a novel hybrid quantum deep neural network(HQDNN) used for image classification. After bilinear interpolation reduces the original image to a suitable size, an improved novel enhanced quantum representation(INEQR) is used to encode it into quantum states as the input of the HQDNN. Multi-layer parameterized quantum circuits are used as the main structure to implement feature extraction and classification. The output results of parameterized quantum circuits are converted into classical data through quantum measurements and then optimized on a classical computer. To verify the performance of the HQDNN, we conduct binary classification and three classification experiments on the MNIST(Modified National Institute of Standards and Technology) data set. In the first binary classification, the accuracy of 0 and 4 exceeds98%. Then we compare the performance of three classification with other algorithms, the results on two datasets show that the classification accuracy is higher than that of quantum deep neural network and general quantum convolutional neural network.展开更多
For the two-level atoms system interacting with single-mode active field in a quantum cavity, the dynamics of the Bose-Einstein Condensation (BEC) is analyzed using an ordinary method suggested by authors to solve the...For the two-level atoms system interacting with single-mode active field in a quantum cavity, the dynamics of the Bose-Einstein Condensation (BEC) is analyzed using an ordinary method suggested by authors to solve the system of Schrodinger representation in the Heisenberg representation. The wave function of the atoms is given. The stability factor determining the BEC and the selection rules of the quantum transition are solved.展开更多
We reveal that the two-variable Hermite function hm,n, which is the generalized Bargmann representation of the two-mode Fock state, involves quantum entanglement of harmonic oscillator＇s wave functions. The Schmidt d...We reveal that the two-variable Hermite function hm,n, which is the generalized Bargmann representation of the two-mode Fock state, involves quantum entanglement of harmonic oscillator＇s wave functions. The Schmidt decomposition of hm,n is derived. It also turns out that hm,n can be generated by windowed Fourier transform of the single-variable Hermite functions. As an application, the wave function of the two-variable Hermite polynomial state S（γ）Hm,n （μa1^＋, μa2^＋│00〉, which is the minimum uncertainty state for sum squeezing, in （ η│representation is calculated.展开更多
文摘We mostly investigate two schemes. One is to teleport a multi-mode W-type entangled coherent state using a peculiar bipartite entangled state as the quantum channel different from other proposals. Based on our formalism,teleporting multi-mode coherent state or squeezed state is also possible. Another is that the tripartite entangled state is used as the quantum channel of controlled teleportation of an arbitrary and unknown continuous variable in the case of three participators.
基金National Natural Science Foundation of China under Grant No.10775097
文摘Based on our previous paper (Commun.Theor.Phys.39 (2003) 417) we derive the convolution theoremof fractional Fourier transformation in the context of quantum mechanics,which seems a convenient and neat way.Generalization of this method to the complex fractional Fourier transformation case is also possible.
基金Supported by Doctoral Foundation of Qingdao University of Science and Technology (20080022398)the National Natural Science Foundation of China (11271318, 11171296)the Specialized Research Fund for the Doctoral Program of Higher Education of China (20110101110010)
文摘For a field k and two finite groups G and X, when G acts on X from the right by group automorphisms, there is a Hopf algebra structure on k-space （kX°P）＊ kG （see Theorem 2.1）, called a non-balanced quantum double and denoted by Dx（G）. In this paper, some Hopf algebra properties of Dx （G） are given, the representation types of Dx （G） viewed as a k-algebra are discussed, the algebra structure and module category over Dx（G） are studied. Since the Hopf algebra structure of non-balanced quantum double DX （G） generMizes the usual quantum double D（G） for a finite group G, all results about Dx（G） in this paper can also be used to describe D（G） as a special case and the universal R-matrix of Dx （G） provides more solutions of Yang-Baxter equation.
基金supported by the National Key Research and Development Program of China(Grants No.2017YFA0304202 and No.2017YFA0205700)the NSFC(Grants No.11875231 and No.11935012)the Fundamental Research Funds for the Central Universities through Grant No.2018FZA3005。
文摘Majorana's stellar representation provides an intuitive picture in which quantum states in highdimensional Hilbert space can be observed using the trajectory of Majorana stars.We consider the Majorana's stellar representation of the quantum geometric tensor for a spin state up to spin-3/2.The real and imaginary parts of the quantum geometric tensor,corresponding to the quantum metric tensor and Berry curvature,are therefore obtained in terms of the Majorana stars.Moreover,we work out the expressions of quantum geometric tensor for arbitrary spin in some important cases.Our results will benefit the comprehension of the quantum geometric tensor and provide interesting relations between the quantum geometric tensor and Majorana's stars.
基金Supported by National Natural Science Foundation of China under Grant No.11031005Beijing Municipal Education Commission Foundation under Grant Nos.KZ201210028032 and KM201210028006
文摘We construct a fermion analogue of the Fock representation of quantum toroidal algebra and construct the fermion representation of quantum toroidal algebra on the K-theory of Hilbert scheme.
文摘To consummate the quantum pendulum theory whose Hamiltonian takes bosonic operator formalism and manifestly exhibits its dynamic behaviour in the entangled state representation, we introduce angular momentum state representation and phase state representation. It turns out that the angular momentum state is the partial wave expansion of the entangled state.
基金The project was supported by the Natural Science Foundation of Fujian Province of China(No.S0750012)
文摘In this paper,we present three types of representations over Anq defined based on the quantum torus of rank n,which are closely related to modules over some vertex algebras.The isomorphism classes among these modules are also determined.
基金Project supported by the Natural Science Foundation of Heze University of Shandong Province of China (Grant Nos XY07WL01 and XY05WL01)the University Experimental Technology Foundation of Shandong Province of China (Grant No S04W138)the National Natural Science Foundation of China (Grant No 10574060)
文摘This paper constructs the new common eigenvectors of n intermediate coordinate-momentum operators which are complete and orthonormal. The intermediate coordinate-momentum representation of a multi-particles system is proposed and applied to a generaln-mode quantum harmonic oscillators system with coordinate-momentum coupling.
基金supported by the National Natural Science Foundation of China(Grant Nos.12001480 and 11871318)the Applied Basic Research Program of Shanxi Province(Grant Nos.201901D211461 and 201901D211462)+2 种基金the Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi(Grant No.2020L0554)the Excellent Doctoral Research Project of Shanxi Province(Grant No.QZX-2020001)the PhD Start-up Project of Yuncheng University(Grant No.YQ-2019021)。
文摘With the rapid development of machine learning,artificial neural networks provide a powerful tool to represent or approximate many-body quantum states.It was proved that every graph state can be generated by a neural network.Here,we introduce digraph states and explore their neural network representations(NNRs).Based on some discussions about digraph states and neural network quantum states(NNQSs),we construct explicitly an NNR for any digraph state,implying every digraph state is an NNQS.The obtained results will provide a theoretical foundation for solving the quantum manybody problem with machine learning method whenever the wave-function is known as an unknown digraph state or it can be approximated by digraph states.
基金Supported by the National Natural Science Foundation of China(Nos.12001480,11871318)the Applied Basic Research Program of Shanxi Province(No.201901D211461)+2 种基金the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi(No.2020L0554)the Excellent Doctoral Research Project of Shanxi Province(No.QZX-2020001)the PhD Start-up Project of Yuncheng University(No.YQ-2019021)。
文摘The quantum many-body problem(QMBP) has become a hot topic in high-energy physics and condensed-matter physics. With an exponential increase in the dimensions of Hilbert space, it becomes very challenging to solve the QMBP, even with the most powerful computers. With the rapid development of machine learning, artificial neural networks provide a powerful tool that can represent or approximate quantum many-body states. In this paper, we aim to explicitly construct the neural network representations of hypergraph states. We construct the neural network representations for any k-uniform hypergraph state and any hypergraph state,respectively, without stochastic optimization of the network parameters. Our method constructively shows that all hypergraph states can be represented precisely by the appropriate neural networks introduced in [Science 355(2017) 602] and formulated in [Sci. China-Phys.Mech. Astron. 63(2020) 210312].
基金supported by the National Natural Science Foundation of China(Grant Nos.11964013,11664017)the Training Program for Academic and Technical Leaders of Major Disciplines in Jiangxi Province(20204BCJL22053)。
文摘A beam splitter operator is a very important linear device in the field of quantum optics and quantum information.It can not only be used to prepare complete representations of quantum mechanics,entangled state representation,but it can also be used to simulate the dissipative environment of quantum systems.In this paper,by combining the transform relation of the beam splitter operator and the technique of integration within the product of the operator,we present the coherent state representation of the operator and the corresponding normal ordering form.Based on this,we consider the applications of the coherent state representation of the beam splitter operator,such as deriving some operator identities and entangled state representation preparation with continuous-discrete variables.Furthermore,we extend our investigation to two single and two-mode cascaded beam splitter operators,giving the corresponding coherent state representation and its normal ordering form.In addition,the application of a beam splitter to prepare entangled states in quantum teleportation is further investigated,and the fidelity is discussed.The above results provide good theoretical value in the fields of quantum optics and quantum information.
基金supported by the National Key Research and Development Program of China(Grant No.2023YFA1009801)National Natural Science Foundation of China(Grant No.12271529)。
文摘We provide a direct proof of the Seidel representation in the quantum K-theory QK(Gr(k,n))by studying projected Gromov-Witten varieties concretely.As applications,we give an alternative proof of the K-theoretic quantum Pieri rule by Buch and Mihalcea(2011),reduce certain quantum Schubert structure constants of higher degree to classical Littlewood-Richardson coefficients for K(Gr(k,n)),and provide a quantum Littlewood-Richardson rule for QK(Gr(3,n)).
基金the National Natural Science Foundation of China (Grant No. 6217070290)Shanghai Science and Technology Project (Grant Nos. 21JC1402800 and 20040501500)。
文摘As a part of quantum image processing, quantum image scaling is a significant technology for the development of quantum computation. At present, most of the quantum image scaling schemes are based on grayscale images, with relatively little processing for color images. This paper proposes a quantum color image scaling scheme based on bilinear interpolation, which realizes the 2^(n_(1)) × 2^(n_(2)) quantum color image scaling. Firstly, the improved novel quantum representation of color digital images(INCQI) is employed to represent a 2^(n_(1)) × 2^(n_(2)) quantum color image, and the bilinear interpolation method for calculating pixel values of the interpolated image is presented. Then the quantum color image scaling-up and scaling-down circuits are designed by utilizing a series of quantum modules, and the complexity of the circuits is analyzed.Finally, the experimental simulation results of MATLAB based on the classical computer are given. The ultimate results demonstrate that the complexities of the scaling-up and scaling-down schemes are quadratic and linear, respectively, which are much lower than the cubic function and exponential function of other bilinear interpolation schemes.
文摘We use the Ringel-Hall algebra approach to study the canonical basis elements for the quantum group of type B2 which are characterized in Xi [12]. However, our approach simplifies several computations there.
基金supported by the Natural Science Foundation of Hunan Province of China (Grant No. 10JJ3088)the Key Research Foundation of the Education Bureau of Hunan Province of China (Grant No. 08A015)the Funds of the Hunan Education Bureau of China (Grant Nos. 10C0616 and 08C344)
文摘The entanglement dynamics of two-qubit systems in different quantum noises are investigated by means of the operator-sum representation method. We find that, except for the amplitude damping and phase damping quantum noise, the sudden death of entanglement is always observed in different two-qubit systems with generalized amplitude damping and depolarizing quantum noise.
文摘Let H be a finite Hopf C^＊ -algebra and H′be its dual Hopf algebra. Drinfeld＇s quantum double D（H） of H is a Hopf^＊-algebra. There is a faithful positive linear functional θ on D（H）. Through the associated Gelfand-Naimark-Segal （GNS） representation, D（H） has a faithful^＊ -representation so that it becomes a Hopf C^＊ -algebra. The canonical embedding map of H into D（H） is isometric.
基金supported by the National Natural Science Foundation of China(No.21373191,No.21573202,No.21633006,and No.21703225)the Fundamental Research Funds for the Central Universities(No.2030020028,No.2060030025,and No.2340000074)
文摘We present a semiclassical （SC） approach for quantum dissipative dynamics, constructed on basis of the hierarchical-equation-of-motion （HEOM） formalism. The dynamical components considered in the developed SC-HEOM are wavepackets＇ phase-space moments of not only the primary reduced system density operator but also the auxiliary density operators （ADOs） of HEOM. It is a highly numerically efficient method, meanwhile taking into account the high-order effcts of system-bath couplings. The SC-HEOM methodology is exemplified in this work on the hierarchical quantum master equation [J. Chem. Phys. 131, 214111 （2009）] and numerically demonstrated on linear spectra of anharmonic oscillators.
基金Project supported by the Natural Science Foundation of Shandong Province,China (Grant No. ZR2021MF049)the Joint Fund of Natural Science Foundation of Shandong Province (Grant Nos. ZR2022LLZ012 and ZR2021LLZ001)。
文摘We redesign the parameterized quantum circuit in the quantum deep neural network, construct a three-layer structure as the hidden layer, and then use classical optimization algorithms to train the parameterized quantum circuit, thereby propose a novel hybrid quantum deep neural network(HQDNN) used for image classification. After bilinear interpolation reduces the original image to a suitable size, an improved novel enhanced quantum representation(INEQR) is used to encode it into quantum states as the input of the HQDNN. Multi-layer parameterized quantum circuits are used as the main structure to implement feature extraction and classification. The output results of parameterized quantum circuits are converted into classical data through quantum measurements and then optimized on a classical computer. To verify the performance of the HQDNN, we conduct binary classification and three classification experiments on the MNIST(Modified National Institute of Standards and Technology) data set. In the first binary classification, the accuracy of 0 and 4 exceeds98%. Then we compare the performance of three classification with other algorithms, the results on two datasets show that the classification accuracy is higher than that of quantum deep neural network and general quantum convolutional neural network.
文摘For the two-level atoms system interacting with single-mode active field in a quantum cavity, the dynamics of the Bose-Einstein Condensation (BEC) is analyzed using an ordinary method suggested by authors to solve the system of Schrodinger representation in the Heisenberg representation. The wave function of the atoms is given. The stability factor determining the BEC and the selection rules of the quantum transition are solved.
文摘We reveal that the two-variable Hermite function hm,n, which is the generalized Bargmann representation of the two-mode Fock state, involves quantum entanglement of harmonic oscillator＇s wave functions. The Schmidt decomposition of hm,n is derived. It also turns out that hm,n can be generated by windowed Fourier transform of the single-variable Hermite functions. As an application, the wave function of the two-variable Hermite polynomial state S（γ）Hm,n （μa1^＋, μa2^＋│00〉, which is the minimum uncertainty state for sum squeezing, in （ η│representation is calculated.
