Now that we know how to represent our state vector as a superposition of states, and we know that we can only measure the state vector to be in one of the base states, it would seem that there would be some sort of discrepancy. We must determine what happens when we measure the state vector. We know from quantum physics that given an initial condition the state vector will evolve in time in accordance with Schrödenger's equation:
In our notation this expression is:
Now understanding that a state vector will collapse when it interacts with the external environment, we still need to know in what manner this collapse happens. To perform any sort of useful calculation we must be able so say something about which base state a quantum mechanical system will collapse into. The probability that the state vector will collapse into the j'th eigenstate, is given by | wj|2 which is defined to be aj2 + bj2 if wj = aj + i*bj, where wj is the complex projection of the state vector onto the j'th eigenstate. In general the chance of choosing any given state is
So now we know how to construct an n state quantum system, which can be placed in an arbitrary superposition of states. We also know how to measure the resultant superposition and get a base state with a certain probability. This is all that we need to understand about our quantum memory register to be able to simulate its behavior.