Applications Of Thermo Entangled State Representation In Solving Some Quantum Master Equations

The problem of decoherence is an integral part of the theory of quantum computation and communication. The potential of a quantum computer lies in its ability to process information in the form of a coherent superposition of quantum mechanical states. Because quantum coherence and interference play a central role in a quantum computer, decoherence is a major threat to its proper functioning. Asimilar situation prevails in quantum communication. The central problem of quantum communication is how to faithfully transmit unknown quantum states through a noisy quantum channel. While quantum information is sent through a channel such as an optical fiber, the carriers of the information (e.g. photons) interact with the channel and get entangled with its many degrees of freedom, which gives rise to the phenomenon of decoherence on the state space of the information carriers. An initially pure state becomes a mixed state when it leaves the channel. One of the major topics in Quantum Statistical Mechanics is the evolution of pure states into mixed states. Such evolution usually happens when a system is immersed in a thermal environment, or a signal (a quantum state) passes through a quantum channel, and is described by a master equation. Master equations are set up for a better understanding of how quantum decoherence is processed to affect unitary character in the dissipation or gain of the system. On the other hand, according to the quantum information theory the decay (decoherence) due to the interaction between a system and its environment can be described by a superoperator.Usually, as shown in the literature before, solving master equations is using either the Langevin equation or the Fokker-Planck equation after recasting the density operators into some definite representations, e.g. particle number representation (Q-function), coherent state representation (P- representation) or the Wigner representation. In this thesis we alternatively treat this equation by virtue of the newly developed thermo entangled state representation. Because it is recently acknowledged that there involves quantum entanglement during the evolution between the system and its environment. Thus quantum entanglement should be taken into account in treating decoherence. We construct the appropriate entangled state representation to tackle the evolution issue from pure states to mixed states.First we show that passing through the amplitude dissipative channel the initial pure number state density operator is evolved into the density operator of binomial distribution (a mixed state), and the binomial distribution parameter is just equal to e 2kt, where k is the dissipative parameter of the channel. We solve the corresponding master equation to obtain the operator-sum representation of density operator by virtue of the entangled state representation, which seems a convenient approach.Second we investigate how the density operator of a squeezed chaotic field (a mixed state) evolves in the amplitude dissipative channel. We demonstrate that the evolved density operator t is in a Gaussian quadratic form, and the Q-function of tis derived, which manifestly exhibits the dissipation.Third, we solve the newly constructed nonlinear master equation 21(?) is an operator-valued function of N a a , for describing amplitude damping channel, and derive the infinite operator sum representation of quasi-Kraus operators for the density operator. We also show that in this nonlinear process the initial pure number state density operator will evolve into the binomial field (a mixed state) when 1f N N 1.Fourth, we solve a kind of newly constructed two-mode coupled master equation for describing amplitude damping channel, and derive the infinite operator sum representation of Kraus operators for the density operator.Fifth we review and summarize the methods of solving the master equations by the thermal entangled state representation, in order to find the solution more general rules. We hope this methods may have deeper development in future.In summary, we have adopted the entangled state approach for treating the time evolution of the density operator in various quantum channels. The result helps us to grasp the inward nature (quantum entanglement) of decoherence in an intuitive manner. Using the entangled state representation, we can also extend the discrete sum of operators to the continuous sum of operators for describing quantum channels. Throughout our discussions, we shall make full use of the technique of integration within an ordered product (IWOP) of operators to deal with integrations over ket-bra operators.