Often we have incomplete information about a system, we do not know the initial state of the system perfectly well.
In general we handle incomplete information about the initial state using the concept of probability. We can say that the system has a probability pk of being in a state |ψk>. (∑kpk = 1.)
We say that we are dealing with a statistical mixture of states. We now want to know the probability of measuring the eigenvalue an if a measurement of the observable A is made.
Note: Probabilities enter at two levels.
A statistical mixture of states is NOT equivalent to a linear superposition of states. If a system is in a state ∑kckψk, then its initial state is exactly known. If the system has probability pk of being in the state |ψk>, then its initial state is not exactly known. Even if pk = ck the probability of obtaining the eigenvalue an when measuring A is in general not the same for the two systems. Interference effects are absent for a statistical mixture. We cannot describe a statistical mixture using an "average state vector". However an "average operator", called the density operator permits a simple description of a statistical mixture.