Optimization of the rotation angle of a quantum particle in a simplequantum system

In this text we used a quantum system of two quantum particles, specificallytwo qubits. One qubit acts as a detector in an equal weight superposition ofspin up and spin down. The two qubits in the system are entangled. To domultiple measurements on the system, we want to rotate the system back to theequal weight superposition, which is destroyed after a measurement.We have calculated the time evolution of the density matrix of this quantumsystem using a standard differential equation. We altered this differential equa-tion using the counting fields method. After this we used the density matrix tocalculate the optimal angle of rotation to return the qubit to the equal weightsuperposition. We did this for an ideal system without outside influences andfor a system with outside influences. We derived a relationship between theoutside influences on the system,γ, whereγ= 2 represents an ideal system,and a variable inside the system,z. This relation is2γ=z.We also described an algorithm to simulate a random quantum trajectory.We found that the value of the spin in the z-direction does go to the expectedvalue of 0 when the amount of single trajectories in the average trajectory islarge. This is true for all values ofγ. For the value of the spin in the x-directionwe found that the theoretical curve correctly predicts the behavior of an idealsystem. We found that for a non ideal system, whenγ >2, the theoreticalcurve does not correctly predict the behavior of a trajectory. The theory fails topredict the speed with which the spin in the x-direction goes to 0 in a non idealsystem, where the theory is significantly slower at going to 0. Also the theorypredicts the starting value to be lower than 1 in a non ideal system, which isnot the case in practic