Preparation Of Quantum Logic Gate, Generation Of Entanglement States And Quantification Of Thermal Entanglement

The development of quantum information is due to the combination of quantum theory and information theory. The investigation of quantum computer and quantum teleportation have been attracted much attention in recent years. One of the advantages of quantum computer is quantum parallel calculation. The quantum computer can perform ultra-fast calculation, simulate quantum system, and solve problems that cannot be solved by classical digital computer. In this paper, the realization of quantum logic gate in trapped ion, the preparation of multi-qubit entanglement state and the measure of mixed entanglement states in spin-1/2 Heisenberg system are discussed. Quantum computer is formed through combination of quantum logic gates. Since any logic gate can be constructed by two-qubit controlled not gates (CN gate) and one qubit rotation gate (R gate), the realization of the CN and R gates has been studied extensively. We have proposed a simple method to realize CN gate in an ion trap system. The internal state of the ion is a target qubit. The internal ground state |g> and excited state |e> can be expressed by |0> and |1> respectively. The phonon state |0> and |1> are used as control bits. Using two laser beams that are perpendicular to each other, the ion in the trap can be controlled. The coupling strengths along x and y axis can be changed. If the evolution time is properly chosen when the Lamb-Dicke (LD) parameter and the coupling coefficient satisfy certain relation, the CN gate can be realized in the subspace {|00> ,|01>,|10>,|11>}. In this scheme, there is no limitation of LD parameter and no auxiliary level required. Only a two-level ion is necessary. It can be easily realized in an experiment. In quantum computation, the quantum state is the carrier of the information. The manipulation of the quantum information is the process of controlling quantum states. The control of multi-particle entanglement state is the basis of quantum computation. We propose a method to prepare multi-particle entanglement W and GHZ states in cavity QED. The scheme is as follows. The atom is in the state of superposition of 0 and 1 . The cavity is in the coherent state. The atom and the cavity interact with each other with proper interaction time. Then the cavity is measured. If the measured cavity is in different states, the corresponding state of the atom collapses to different entanglement state. It is proven that the prepared entanglement state violates the Bell's inequality. These states have the property of non-locality. If the prepared states satisfy the Bell's inequality, the measured expectation value is calculated. For four-particle W state, the relation is contradictory when the parameters are different. For four-particle GHZ state, the mathematical expression is in contradiction with the physical parameters. It is obvious that the states violate the Bell's inequality. The quantum entanglement is an important resource of quantum computation and quantum information. Through entanglement state, the quantum teleportation, quantum coding and quantum computation can be realized. The Bell's inequality is the criterion of the entanglement state, but it cannot be used to calculate the entanglement quantitatively. Therefore, scientists are seeking methods to measure the entanglement. For two-particle pure and mixed states, the measurement of the entanglement is well developed. The entanglement of multi-particle pure state can also be calculated. However, the entanglement measurement of multi-particle mixed states still needs to be developed. We investigate the thermal entanglement of two and multi particles in two-and three-dimensional Heisenberg models. Through the quantity of concurrence, the entanglement can be calculated analytically. Through the extended theory of n-concurrence, the global entanglement of three-dimensional Heisenberg XY model in the form of tetrahedron and cubic lattices is investigated. It is shown that the concurrence and global entanglement are the functions of the temperature, the coupling strength and the external magnetic field. One can control the concurrence and the global entanglement through changing the parameters of the system.