Construction And Applications Of New Continuous Variable Entangled State Representation

Quantum entanglement reflects the correlation and the inseparability between the variousparts of the two-body and many-body quantum systems, which become an investigated hot spotin the current field of quantum optics, quantum computing, quantum information, etc., With thedeepening of research on quantum entanglement, the basic and important understanding of thepeople in quantum information science becomes more and more profound. Quantumentanglement is a very important physical resource which closely related with quantuminformation. The purpose of the study of quantum entanglement problem is the development andapplication of this new physical resource. For example, when the two parts share a certainamount of entanglement, the functions of quantum communication and quantum computing canbe manipulated, such as quantum key distribution, quantum dense coding, remote quantumcomputing, etc. For a specific dynamics system problem in quantum mechanics, finding asuitable representation can greatly solve the problem and reduce the workload. The phenomenonof quantum entanglement is only obviously expressed in the entangled state representation.Therefore, the discovery of more entangled state representations and study it’s applications havevery important scientific value and broad application prospects. The main work of this paper isto establish a series of representations of the entanglement state, intermediate entangled state,coherent entangled state, etc. The entangled state representations can be extended to the moregeneral case with generalized entangled state representations, which have very importantscientific value and broad application prospects.The paper is divided into seven chapters, a total of four parts, which cover the basicrepresentations and transformations of quantum mechanics, construction and applications of theentangled state representations, construction and applications of the intermediary entangled staterepresentations, construction and applications of coherent entangled state representations. Thespecific arrangements are as follows:The first part (the first chapter) includes two aspects. First, we will introduce the coordinaterepresentation, momentum representation, coherent state representation, two-mode continuousvariable entangled state representation in the field of quantum mechanics and quantum optics,and re-examine completeness of these basic representation with the normal product. Second, theunitary quantum squeezed transform operators and rotational transformation operators are constructed by the technique of integration within ordered product and based on asymmetricket-bra integration with Dirac notation in quantum mechanics.The second part consists of the second chapter, the third chapter and chapter VI. Ageneralized two-mode continuous variable entangled state with real parameters in the secondchapter is proposed and spans a complete and orthonormal representation. The application of thisentangled state is discussed in quantum teleportation. The third chapter gives a newrepresentation of the quantum entangled states about the configurations space rotation, and theHadamard transform operator, two-mode squeezed operator are constructed. The properties andapplications of these operators are discussed. The chapter VI deals with two types of multimodeentangled states. First, a new multimode entangled state representation is proposed by thecommon eigenstate of the commutative operators. Second, ageneralized multimode entangled state representation with ir e1al parameters is proposed by unitarytransformation matrix. The corresponding conjugate state, multi-mode squeezing operator andmultimode Wigner operator are constructed. The properties and applications of these operatorsare discussed.The fourth chapter of the content is the third part of this article, mainly related to the twotypes of intermediary entangled states. First, a new intermediate entangled state representation isproposed by the two-mode coordinate representationq1,q2and two-mode momentumrepresentationp1,p2. Second, a new intermediate entangled state representation is proposedby common eigenstate of the commutative operatorsQ1Q2andP1P2, and a new mutuallyconjugating intermediate entangled state representation is proposed by common eigenstate of thecommutative operatorsQ1Q2andP2P1. Furthermore, the generalized squeezedtransformation operator, Fresnel transformation operator and Hadamard transformation operatorare constructed. A new convenient formalism of quantum tomogram is thus established by thegeneralized coordinates-momentum intermediary entangled state representation1,2,The part IV consists of the chapter V and chapter VII. A coherent entangled staterepresentation in the chapter V is proposed by common eigenstate q, of the commutativeoperatorsQ1Q2anda1a2, which can be produced by using a beam splitter. Thegeneralized Fresnel operator and Hadamard-Fresnel combination operator are constructed by thiscoherent entangled state. On this basis, this two-mode entangled coherent state extended to themore general case, a generalized two-mode continuous variable entangled state representationq,; with real parameters is proposed. The generalized P-representation and some complexoperators are derived by virtue of the normal product with this entangled state. A multimodecoherent entangled state representation in the chapter VII is proposed by common eigenstate ofthe commutative operators, which can be produced by using a beam splitter. The generalized multimode squeezed transformation operator is constructed by asymmetric ket-braintegration transformation. The properties of this squeezed operator are discussed.