Often we have incomplete information about a system, we do not know the initial state of the system perfectly well.

Example:

• Two beams of linearly polarized Laser light with the same frequency are polarized perpendicular to each other and are combined on a target.  We do not know the polarization of an individual photon striking the target, since our detector can only count photons.  We only know that the probability that the photon has one or the other polarization, since we know the intensities of the two beams.  We have incomplete information about the individual photons.

In general we handle incomplete information about the initial state using the concept of probability.  We can say that the system has a probability p_k of being in a state |ψ_k>.  (∑_kp_k = 1.)

We say that we are dealing with a statistical mixture of states.  We now want to know the probability of measuring the eigenvalue a_n if a measurement of the observable A is made.

• The initial information about the system is given in terms of probabilities.
• The predictions of Quantum Mechanics are probabilistic.

A statistical mixture of states is NOT equivalent to a linear superposition of states.  If a system is in a state  ∑_kc_kψ_k, then its initial state is exactly known.  If the system has probability p_k of being in the state |ψ_k>, then its initial state is not exactly known.  Even if p_k = c_k the probability of obtaining the eigenvalue a_n 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.