Applications Of Continuous Variable Entangled State Representation In The Quantum Measurement And In The Thermo Field Dynamics

The quantum entanglement is a core resource in quantum information science. If the quantum entanglement can be represented by the entangled state representation, the quantum entanglement can be more clearly expounded. The establishment of continuous variable entangled state representation can bring great convenience to quantum measurement. Since most realistic systems are immersed in a"thermal reservoir", there is a certain quantum entanglement in the total system of the subsystem and the reservoir. Some applications of continuous variable entangled state representation in the quantum measurement and in the thermo field dynamics (TFD) theory are the major discussions of the paper.We calculate the result of the quadrature-amplitude measurement on the first-mode field of a two-mode squeezed vacuum state by virtue of the entangled state representation. It is found that the second-mode field collapses to a single-mode squeezed state with stronger squeezing. The explicit form is derived.By virtue of the entangled state representation, we analyze the result of one- mode quadrature-amplitude measurement for some generalized two-mode squeezed states with two squeezing parameters. It is found that the remaining field-mode simultaneously collapses to the single-mode squeezed state, which is related to both squeezing parameters. When a measurement for the 3-parameter SU(1,1) squeezing state is performed with the first-mode field in the single-mode squeezed state, the second-mode field becomes a single-mode squeezed state. Its squeezing depends not only on the SU(1,1) parameter, but also on the squeezing parameter of the measurement operator. This shows the quantum entanglement with squeezing. That is, the light field mode can be controlled by the measurement mode, so as to establish a new quantum state.Based on the fact that the two-mode squeezed number state is a two-variable Hermite polynomial excitation of the two-mode squeezed vacuum state, the result of one-mode l-photon measurement for the two-mode squeezed number states S 2 m, n is obtained. It is found that the remaining field-mode simultaneously collapses to the number state n ? m + l with the coefficient being a Jacobi polynomial of n ,m and l . It manifestly exhibits the entanglement between the two modes, i. e., it depends on the number-difference between the two modes. The second mode collapses to an excited coherent state when the first mode is measured as a coherent state.Therefore, the quantum teleportation theory can be further developed by these results.By analyzing the characteristics of thermo field dynamics, a new physical state |z,N> at thermal equilibrium has been found. It is the thermo-invariant coherent state. This state not only maintains the whole energy of the system and the reservoir but also is a generalized coherent state. When the whole system is in this state, the energy of the whole system will not change by simultaneously annihilating a quantum of the system and a hole with negative energy in the reservoir.By using thermal Winger operator of the TFD in the coherent thermal state representation and the integration technique in an ordered product of operators, the Wigner function of the thermo-invariant coherent state |z,N> is derived. The nonclassical property of the state |z,N> is discussed based on the negativity of the Wigner function. It is found that the nonclassical statistical properties of the thermo- invariant coherent state |z,N> depend on both the parameter z of the coherent state and the energy N of the whole system in thermal equilibrium. For different values of N, the state |z,N> can exhibit different nonclassical statistical properties. The nonclassicality of the state |z,N> is more pronounced when N is an odd number.In the thermo field dynamics, every real optical field mmode can be accompanied by a fictitious field mode. Therefore, the photon number state (n-excitation state) can be denoted by |n,(n|ˇ)> . The Hamiltonian of the total system including the effects of the reservoir holds for h |n,(n|ˇ)> = 0. It cannot embody the eigenvalue n. It is inconvenient to describe thermal excitation of the system at finite temperature. To avoid the weakness of |n,(n|ˇ)> , new thermo excitation states are successfully constructed by introducing appropriate thermo excitation operators. It is the eigenvector of the Hamiltonian h with the eigenvalue D. The state ||D ,n> can exhibit not only the eigenvalue D of h, but also the thermal excitation behaviour. It is noted that it is a thermal excitation state and an entangled state. Applications of the new thermo excitation state in constructing new thermo squeezed state and phase state are also presented. The TFD theory can be further developed by the thermo- invariant coherent state and the new thermo excitation state.