Continuous Entangled State Representation: Construction And Its Applications In Wavelet Transformation

Quantum entanglement,which was firstly proposed in 1935 by Einstein-Podolsky-Rosen(EPR) and was further elaborated by Schr(o|¨)dinger,is a special concept of quantum mechanics,it reflects the quantum correlation between the various parts of bipartite system or multipartite system and shows the mysterious and unknown side of quantum mechanics.Since Bennett et al proposed the scheme of quantum teleportation in 1993,there has been much progress in quantum information. In the vast majority of quantum information scheme,such as quantum teleportation, quantum cloning,dense coding,quantum error correction,quantum algorithm,etc., quantum entanglement is one of the key features.Thus,entanglement is a basic issue of quantum information theory.As we know quantum entanglement is concretely embodied in quantum entangled states,so quantum entangled states have become an indispensable subject for quantum information research.In addition,many entangled issues and dynamical problems can only be clearly explained and solved in entangled state representations,and proper representations will bring more convenience in dealing with various problems.From the above discussion we know that finding more useful entangled state representations is inevitable for physicists.But,a question naturally arises:how to establish these entangled state representations? Can we present a general approach to it? In this thesis we will give an affirmative reply.We mainly do the following two aspects research:一,With the help of method of integration within an ordered product(IWOP) of operators and the normally ordered Gaussian operator in quantum mechanical completeness relation we present a new approach for finding many new quantum mechanical representations,and show its application in constructing Wigner operators for multipartite entangled states.In this way,we enriched Dirac's representation and transformation theory and developed the Wigner phase-space distribution function.二,We introduce the Dirac's symbolic method into wavelet transform research. Using the newly constructed coherent-entangled representation|a,x>,we found new unitary operator,and the transformation matrix element of this unitary operator in the entangled state representation |η> is just the mixed Lens-Fresnel transform in classical optics.Then,through the combination of dilation transform and symplectic transrom we construct new mother wavelet and present new wavelet transform,which enriched and developed the theory of wavelet analysis and its application research. The main content of this paper is arranged as follows:In chapterⅠ,we shall briefly introduce some preliminary knowledge on the technique of integration within an ordered product(IWOP).Here,we firstly introduce the Fock representations of the coordinate eigenvector and the momentum eigenvector, respectively,and review the coherent state representation and its properties.Then we list some properties of normal product of operators and exhibit some applications of the IWOP technique.In chapterⅡ,with the help of the IWOP technique and the normally ordered Gaussian operator in quantum mechanical completeness relation we shall concretely introduce a new approach for finding many new quantum mechanical representations, and show its application in constructing Wigner operators for multipartite entangled states.In chapterⅢ,we shall introduce some primary theory on the analysis of wavelets. Here we make a list of the common wavelets,and introduce the definition and properties of continuous wavelet transform.In chapterⅣ,we shall introduce the Dirac's symbolic method into wavelet transform research and construct the quantum mechanical formalism of wavelet transform by the operators and the representations in quantum mechanics.From the admissibility condition of wavelet transform and the representation and transformation theory in quantum mechanics,we present a new,convenient approach to searching for more qualified mother wavelets.Then,using the operator expression we derive the mixed transform of wavelet transform and other optical transforms.In chapterⅤ,using the representation and transformation theory in quantum mechanics,we shall introduce the classical Fresnel transform and Lens-Fresnel transform into wavelet transform theory,and derive a series of new wavelet transform.