Bell Measurement And Squeeze-swapping Of Entangled State Of Continuum Variables

In the1935, Einstein, Podolsky and Rosen put forward the concept of quan-tum entangle. In their paper, the operators of relative coordinate x1-X2and total momentum P1+P2are commutation. So the entangle state of the two par-ticles can’t be factored into the single particle state of product, and at least two parts. Each one of the parts is two single particle state of product. There are entangled of the two particles, without a alone state of single particle. And they have not the natures of independent particle. Many experiments have shown that there are nonlocality of two particles, for the past few years. The state|η> are the common eigenstates of X1-X2and P1-P2and their expressions in Fock space are firstly provided by Hong-yi Fan. These states can constitute a new quantum representation due to their completeness and orthogonality. In this thesis, we discuss the characteristics of|η> and the relationship with other representations. We mainly discussed the following content:the relations of|η> with two-mode squeezed states and coherent states; how to extend|η> to three photons entan-gled state, how to use the state of entanglement swapping research and two-mode squeezed state after registering n-single mode photons. The aim of the thesis is to further clear the physical meaning of|η>. The content of our paper is:1. We research the Schmidt decompositions of entangled state representa-tion of continuum variables by the technique of the integration within an ordered product. We obtain the Schmidt decomposition of|η> in representations of coor-dinate, momentum and particle number (Fock state). We study the influences of the squeezed operator to entangled state representation by the Schmidt decompo-sition method, and find that the two-mode squeezed operator can cause quantum entanglement. Our results may have a contribution to understand the relations of quantum entanglement and squeezed operator. 2. We study entanglement swapping by using Bell measurement of entangled state representation of continuum variables. We give the realization and results of entanglement swapping and find that entanglement swapping can obtain en-tanglement. Because two-mode squeezed state itself is an entangled state, we study squeeze-swapping of entangled state representation of continuum variables and put forward how to obtain squeezed state by squeeze-swapping. The results may not only have some contributions to potential application of squeezed state in quantum communication but also are beneficial to understand the relations of entangled state and squeezed state.3. We study the situation that the two-mode squeezed state make one mode n-photon count. The results show that the two-mode squeezed state will collapse to a counting operator in the remaining mode with the new smaller quantum efficiency of the detector. The beneficial attempt not only provide a new method and idea about photon count but also reflect the entanglement between the two modes of two-mode squeezed state.4. We research the mutual transformation between the coherent state and quantum entangled state, by considering the important role of them in quantum communication. This work may be important to understand representation of quantum state and help experimenters to find new optical transform.The thesis consists of six parts. In chapter one, we briefly review the related concepts of quantum entanglement and use the technique of the integration within an ordered product to prove that the EPR state of continuum variables is a quan-tum representation. In chapter two, we research the Schmidt decompositions of entangled state representation of continuum variables. And study the influences of the squeezed operator to entangled state representation by the Schmidt decom-position method. In chapter three, we discuss the problems of Bell measurement and squeeze-swapping in terms of the entangled state representation. In chapter four, the situation that the two-mode squeezed state make one mode n-photon count is investigated. In chapter five, we analyze the relations of coherent state and entangled state. In chapter six, the summary and prospect are given.